Matroid Theory - WordPress.com...2017/09/01 · Matroid Theory Travis Dirle May 30, 2017 2 Contents 1 Basics and Deﬁnitions 1 2 Duality 9 3 Minors 11 4 Connectivity 19 5 Graphic

Matroid Theory

2

Matroid Theory

Travis Dirle

May 30, 2017

2

Contents

1 Basics and Definitions 1

2 Duality 9

3 Minors 11

4 Connectivity 19

5 Graphic Matroids 29

6 Finite Geometry 33

7 Representable Matroids 39

8 Constructions 47

9 The Splitter Theorem 51

10 Axiom Tables 53

i

CONTENTS

ii

Chapter 1

Basics and Definitions

Definition 1.0.1. A matroid is a pair (E, I) in which E is a finite set and I is afamily of subsets of E satisfying

(I1) I 6= ∅;(I2) if J ∈ I and I ⊂ J , then I ∈ I;(I3) if I, J ∈ I with |I| < |J |, then there is some element x ∈ J − I with

I ∪ {x} ∈ I.The family I are the independant sets of the matroid.

Definition 1.0.2. A pair (E, I) is a set system if I is a nonempty collection ofsubsets of E that is closed under taking subsets.

Proposition 1.0.3. Let E be a finite set and let I be a family of subsets of E.Then the family I are the independent sets of a matroid if and only if:

(I1’) ∅ ∈ I;(I2) if J ∈ I and I ⊂ J , then I ∈ I;(I3’) if I, J ∈ I with |J | = |I|+ 1, then there is some element x ∈ J− I with

I ∪ {x} ∈ I.

Recall: The maximal subsets in a family are those subsets that are not containedin any other sets in the family, they are maximal with respect to inclusion. Thisis different than maximum.

Definition 1.0.4. If M is a matroid with independent sets I, then B is a basis ofthe matroid M if B is a maximal independent set.

Proposition 1.0.5. The following are always true for the bases B of a matroid:(B1) B 6= ∅;(B2) If B1, B2 ∈ B and B1 ⊂ B2, then B1 = B2;(B3) If B1, B2 ∈ B and x ∈ B1 − B2 then there is an element y ∈ B2 − B1

so that B1 − x ∪ {y} ∈ B. (Weak basis exchange).

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CHAPTER 1. BASICS AND DEFINITIONS

An alternate version of (B2) is (B2’): If B1, B2 ∈ B, then |B1| = |B2|. Also,(B2) is known as clutter, i.e. a family of subsets of a set, no one of which is asubset of another. A stronger version of (B3) is the strong basis exchange:

(B3’) If B1 and B2 are bases with x ∈ B1 − B2, then there is an elementy ∈ B2 −B1 so that both B1 − x ∪ {y} and B2 − y ∪ {x} are bases.

Proposition 1.0.6. Let E be a finite set and let B be a family of subsets of E.The family B satisfies (B1),(B2),(B3) if and only if B satisfies (B1),(B2’),(B3).

Theorem 1.0.7. Let E be a finite set and let B be a family of subsets of Esatisfying (B1),(B2),(B3). Then (E,B) is cryptomorphic to the matroid M =(E, I) and B is the collection of bases of a matroid.

Definition 1.0.8. Let M be a matroid on the ground set E. An isthmus/coloopis an element x ∈ E that is in every basis. A loop is an element x ∈ E that isin no basis, or also e is a circuit of M . Moreover, if f and g are elements of Msuch that fg is a circuit, then f and g are parallel in M .

Definition 1.0.9. LetM be a matroid. If C is dependent, but every proper subsetof C is independent, we call C a circuit in the matroid.

Note that, listing the circuits is often an efficient way to describe a matroid. Theycan have very different sizes (unlike bases).

Proposition 1.0.10. The three properties that the family C of circuits satisfiesare:

(C1) ∅ /∈ C;(C2) if C1, C2 ∈ C and C1 ⊂ C2, then C1 = C2;(C3) if C1, C2 ∈ C with C1 6= C2, and x ∈ C1 ∩ C2, then C3 ⊂ C1 ∪ C2 − x

for some C3 ∈ C. (Circuit elimination).

A stronger version of (C3) is the strong circuit elimination:

(C3’) If C1 and C2 are circuits with y ∈ C1 − C2 and x ∈ C1 ∩ C2, then there isa circuit C3 ⊂ C1 ∪ C2 with y ∈ C3 and x /∈ C3.

We can also write the following:

C = {C ⊂ E : C /∈ I and if I ( C then I ∈ I}.

I = {I ⊂ E : ∀C ∈ C, C 6⊂ I}.

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Theorem 1.0.11. Let E be a finite set and let C be a family of subsets of Esatisfying (C1),(C2),(C3). Then (E, C) is cryptomorphic to the matroid M =(E, I) and C is the collection of circuits of M .

Theorem 1.0.12. Let E be a set and C be a collection of subsets of E satisfying(C1)-(C3). Let I be the collection of subsets of E that contain no member of C.Then (E, I) is a matroid having C as its collection of circuits.

Theorem 1.0.13. Suppose tha I is an independent set in a matroid M and e isan element of M such that I ∪ e is dependent. Then M has a unique circuitcontained in I ∪ e, and this circuit contains e.

We see that B is the collection of maximal subsets of E that contain no memeberof C, while C is the collection of minimal sets that are contained in no memberof B.

Definition 1.0.14. Let M = (E, I) be a matroid and let A ⊂ E. The rank ofA, written r(A), is the size of the largest independent subset of A:

r(A) = maxI⊂A{|I| : I ∈ I}.

The rank of a matroid r(M) is just r(E).

Proposition 1.0.15. Let M be a matroid on the ground set E with rank functionr.

i) An element x ∈ E is a loop if and only if for all A ⊂ E with x /∈ A, wehave r(A ∪ {x}) = r(A).

ii) An element x ∈ E is a coloop if and only if for all A ⊂ E with x /∈ A, wehave r(A ∪ {x}) = r(A) + 1.

Theorem 1.0.16. Let E be a finite set with an integer valued function r definedon subsets of E. Then r is the rank function of a matroid if and only if forA,B ⊂ E:

(r1) 0 ≤ r(A) ≤ |A|;(r2) if A ⊂ B, then r(A) ≤ r(B);(r3) r(A ∪B) + r(A ∩B) ≤ r(A) + r(B).

Proposition 1.0.17. Let M be a matroid with rank function r and suppose thatX ⊂ E. Then

i) X is independent⇔ |X| = r(X);ii) X is a basis⇔ |X| = r(X) = r(M); andiii)X is a circuit⇔X is nonempty and for all x ∈ X , r(X−x) = |X|−1 =

r(X).

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Definition 1.0.18. Let E be the ground set of the matroid M . A subset F ⊂ Eis a flat or closed set if r(F ∪ {x}) > r(F ) for any x /∈ F .

Intuitively, flats are the points, lines, planes and higher-dimensional (hyper)planesof a geometry. Some important things to notice: the loops are in every flat,also, if e is an isthmus and F is a flat, then e /∈ F ⇒ F ∪ {e} is a flat, ande ∈ F ⇒ F − {e} is a flat. ∅ will be a flat precisely when M has no loops. E isalways a flat in any matroid, it is the unique flat of rank r(M).

Proposition 1.0.19. If F1 and F2 are flats in a matroid, then so is F1 ∩ F2.

Proposition 1.0.20. Let F be a flat of a matroid M with x /∈ F . Then r(F ∪{x}) = r(F ) + 1.

Definition 1.0.21. Let E be the ground set of a matroid M . A subset H ⊂ E isa hyperplane if H is a flat of M and if r(H) = r(M)− 1.

Lemma 1.0.22. Let M be a matroid on the ground set E with flats F . LetA ⊂ E. Then

i) There is a unique flat F ∈ F such thata) A ⊂ F , andb) If A ⊂ F ′ for some F ′ ∈ F , then F ⊂ F ′.ii) The flat F from part i, satisfies F =

⋂F ′∈F{F ′ : A ⊂ F ′}.

Definition 1.0.23. Let M be a matroid with flats F and let A ⊂ E. Then theclosure of A, written A, is defined by

A =⋂F∈F

{F : A ⊂ F}.

Or also, cl(A) = {x ∈ E : r(A ∪ x) = r(A)}.

It can be seen that A is just the unique smallest flat containing A. Also, the flatsin a matroid are precisely those sets F with F = F . Also, for A ⊂ E, A ⊂ A

and A = A.

Lemma 1.0.24. If A ⊂ B, then r(A ∪ x)− r(A) ≥ r(B ∪ x)− r(B).

Proposition 1.0.25. We have three local properties of the rank function:(r1’) r(∅) = 0;(r2’) For all A ⊂ E and x ∈ E, we have r(A) ≤ r(A ∪ x) ≤ r(A) + 1;(r3’) For x, y /∈ A, if r(A) = r(A∪x) = r(A∪y), then r(A) = r(A∪x∪y).

Proposition 1.0.26. Let r be the rank function for a matroid. For any A ⊂ Eand x1, . . . , xn ∈ E, we have:

(r3”) If r(A) = r(A ∪ x1) = · · · = r(A ∪ xn), then

r(A ∪ {x1, x2, . . . , xn}) = r(A).

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Theorem 1.0.27. Let E be a finite set with closure operator A 7→ A defined onsubsets of E. Then, the closure operator is the closure operator of a matroid ifand only if for A,B ⊂ E:

(c11) A ⊂ A;(c12) If A ⊂ B, then A ⊂ B;(c13) A = A;(c14) If p ∈ A ∪ q − A, then q ∈ A ∪ p. (Maclane-Steinitz exchange).

Proposition 1.0.28. Let M be a matroid and X ⊂ E. Theni) X is a circuit if and only if X is a minimal non-empty set such that x ∈

X − x for all x ∈ X .ii) X = X ∪ {x : M has a circuit C such that x ∈ C ⊂ X ∪ x}.

Theorem 1.0.29. Let E be a finite set and let F be a family of subsets of E.Then the family F are the flats of a matroid if and only if:

(F1) E ∈ F;(F2) If F1, F2 ∈ F , then F1 ∩ F2 ∈ F;(F3) If F ∈ F and {F1, . . . , Fk} is the set of flats that cover F , then {F1 −

F, . . . , Fk − F} partition E − F .

Proposition 1.0.30. LetM be a matroid on the ground setE. Then the followingare equivalent for a subset H ⊂ E:

1) H is a hyperplane;2) H is a maximal proper flat;3) r(H) = r(E)− 1 and r(H ∪ x) = r(E) for all x /∈ H;4) H is maximal with respect to not containing a basis;5) H = H , and H ∪ x = E for all x /∈ H;6) H is covered by E in the lattice of flats;7) H has a unique cover in the lattice of flats.

Definition 1.0.31. A subset X of E is a spanning set of M if X = E. We alsosay that X spans a subset Y of E if Y ⊂ X .

Proposition 1.0.32. Let M be a matroid and X a subset of E. Theni) X is a spanning set⇔ r(X) = r(M);ii) X is a basis⇔ it is both spanning and independent;iii) X is a basis⇔ it is a minimal spanning set;iv) X is a hyperplane⇔ it is a maximal non-spanning set.

Proposition 1.0.33. Let M be a matroid on the ground set E. Then the lat-tice of flats is coatomic, i.e., every flat is the intersection of some collection ofhyperplanes.

Theorem 1.0.34. Let E be a finite set and let H be a family of subsets of E.Then the familyH are hyperplanes of a matroid if and only if:

(H1) E /∈ H;(H2) If H1, H2 ∈ H and H1 ⊂ H2, then H1 = H2;

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(H3) For all distinct H1, H2 ∈ H and for all x ∈ E, there exists H ∈ H with(H1 ∩H2) ∪ x ⊂ H .

Theorem 1.0.35. Let E be the columns of a matrix A with entries in a field F,and let I be the collection of all subsets ofE that are linearly independent. ThenM = (E, I) is a matroid, that is I satisfies (I1),(I2) and (I3).

Proposition 1.0.36. Let E be the set of column labels of anm×n matrix A overa field F, and let I be the set of subsets X of E for which the multiset of columnslabeld by X is a set and is linearly independent in the vector space V (m,F).Then (E, I) is a matroid.

Definition 1.0.37. The matroid obtained from the matrixAwill be denotedM [A]and is called the vector matroid.

Definition 1.0.38. Let E be the set of edges of a graph G and C be the set ofedge sets of cycles of G. Then C is the set of circuits of a matroid on E. Thematroid derived by a graph is called the cycle matroid ofG and denotedM(G).

Definition 1.0.39. Two matroidsM1 andM2 are isomorphic, writtenM1∼= M2,

if there is a bijection ψ from E(M1) to E(M2) such that, for all X ⊂ E(M1),the set ψ(X) is independent in M2 if and only if X is independent in M1. Wecall such a bijection ψ an isomorphism from M1 to M2.

Definition 1.0.40. A matroid that is isomorphic to the cycle matroid of a graphis called graphic.

All matroids on three or fewer elements are graphic.

Definition 1.0.41. If M is isomorphic to the vector matroid of a matrix D overa field F, then M is representable over F and D is a representation for M .

Definition 1.0.42. For a finite set S and a family of subsets of S, (Aj : j ∈ J)with J = {1, . . . ,m}, a transversal is a subset {e1, . . . , em} of S such thatej ∈ Aj for all j ∈ J and the ei are distinct. If X ⊂ S then X is a partialtransversal of (Aj : j ∈ J) if X is a transversal of (Aj : j ∈ K) for somesubset K of J .

Theorem 1.0.43. LetA be a family (A1, . . . , Am) of subsets of a set S. Let I bethe set of partial transversals of A. Then I is the collection of independent setsof a matroid on S.

Proposition 1.0.44. If ∆ is a bipartite graph with vertex classes S and J , thenthe set of subsets X of S that are matched into J is precisely the set of indepen-dent sets of a transversal matroid.

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Definition 1.0.45. A parallel class of M is a maximal subset X of E(M) suchthat any two distinct members of X are parallel and no member of X is a loop.A parallel class is trivial if it contains just one element. A series class of M isa parallel class of M∗. It is non-trivial if it has at least two elements. Thus twoelements are in series in M if and only if they belong to some non-trivial seriesclass of M .

Definition 1.0.46. If M has no loops and no non-trivial parallel classes, it iscalled a simple matroid, or just a geometry.

Definition 1.0.47. Suppose that E is a set that labels a multiset of elements fromV (m,F). Let I be the collection of subsetsX ofE such thatX labels an affinelyindependent subset of V (m,F). Then (E, I) is called the affine matroid on E.If M is isomorphic to such a matroid, we say M is affine over F.

In general, if M is an affine matroid over R of rank m + 1 where m ≤ 3, thena subset X of E is dependent in M if, in the representation of X by points inRm, there are two identical points, or three collinear points, for coplanar points,or five points in space.

Rank 2 matroid drawing procedure from a matrix:Step 1: Draw the vectors in the plane.Step 2: Draw a line in a ’free’ position (a line that is not parallel to any of our

vectors). Now extend each vector to see where it would hit this free line.Step 3: To get a picture of the column vector dependences corresponding to

this matrix, keep the line and discard the original vectors.

Each column vector will be represented by a point. If three vectors are linearlydependent, then the corresponding three points will be collinear. Usually, linesbetween just two points are not drawn (but they are still there).

Rank 3 matroid drawing procedure from a matrix:Step 1: Draw the vectors in three dimensions.Step 2: Find a plane that is free with respect to these vectors.Step 3: Extend or shrink these vectors (or their negatives) until they meet the

plane. These points will be the picture of your column vector dependences.

For a simple rank 3 matroid (no loops, or doubletons), the empty set is inde-pendent, every point and pair is indep, a triple is indep if and only if the threeare not collinear, and no set with more than three points is indep. If the matroidis non-simple, every element in the cloud is a loop and every pair of multiplepoints is a two element dependent set.

Proposition 1.0.48. Let M be a matroid having a subset X that is both a circuitand a hyperplane. Let B′ = B ∪X . Then B′ is the set of bases of a matroid M ′

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on E, moreover

C(M ′) = (C(M)−X) ∪ {X ∪ e : e ∈ E(M)−X}.

The matroid M ′ is called the relaxation of M .

Proposition 1.0.49. Let E = {t, x1, y1, x2, y2, . . . , xr, yr} for some r ≥ 3. LetC1 = {{t, xi, yi} : 1 ≤ i ≤ r} and C2 = {{xi, yi, xj, yj} : 1 ≤ i < j ≤ r}.Let C3 be a possibly empty, subset of {{z1, . . . , zr} : zi ∈ {xi, yi} for all i} suchthat no two members of C3 have more that r − 2 common elements. Finally letC4 be the collection of all (r+ 1)-elements subsets of E that contain no memberof C1 ∪ C2 ∪ C3.

Then there is a rank-r matroid M on E whose collection C of circuits isC1 ∪ C2 ∪ C3 ∪ C4. The matroid is called a rank-r spike with tip t and legsL1, . . . , Lr where Li = {t, xi, yi} for all i. If C3 is empty, the spike is called arank-r free spike with tip.

Definition 1.0.50. Distinct elements e and f of a matroid M are clones if themap that interchanges e and f but fixes every other element of E is an isomor-phism from M to M .

Definition 1.0.51. Let B be a basis of a matroid M . If e ∈ E − B, then B ∪ econtains a unique circuit, C(e, B), called the fundamental circuit of e withrespect to B. Moreover, e ∈ C(e, B).

Definition 1.0.52. Let m and n be non-negative integers with m ≤ n. Let E bean n-element set and B be the collection of m-element subsets of E. We denotethis matroid by Um,n and call it the uniform matroid of rankm on an n-elementset.

I(Um,n) = {X ( E : |X| ≤ m}

8

Chapter 2

Duality

Definition 2.0.1. LetM be a matroid on the ground setE. Then the dual matroidM∗ is a matroid on E so that

B(M∗) = {E −B : B ∈ B(M)}.

Proposition 2.0.2. We have that r(M) + r(M∗) = |E|.

Proposition 2.0.3. We have that (M∗)∗ = M .

Proposition 2.0.4. Let M be a matroid on E, then we have the following rela-tions between a matroid and its dual respectively:

B is a basis⇔ E −B is a basis;I is independent⇔ E − I is spanning;S is spanning⇔ E − S is independent;C is a circuit⇔ E − C is a hyperplane;H is a hyperplane⇔ E −H is a circuit.

We have that U∗m,n = Un−m,n. Also, bases, circuits, spanning sets etc of M∗ arecalled cobases, cocircuits, cospanning sets etc, of M .

Definition 2.0.5. We call a matroid M self-dual if M ∼= M∗ and identicallyself-dual if M = M∗.

Theorem 2.0.6. Let M be a matroid on E with rank function r, and let r∗(A)be the rank of the set A in M∗. Then

r∗(A) = r(E − A) + |A| − r(M).

Proposition 2.0.7. In a matroid M , let C be a circuit and C∗ a cocircuit. Then|C ∩ C∗| 6= 1.

9

CHAPTER 2. DUALITY

Definition 2.0.8. If A is a clutter of subsets of a set S, then the blocker b(A) ofA consists of those minimal subsets of S that have nonempty intersection withevery member of A. Also, b(A) is a clutter.

Proposition 2.0.9. C∗(M) = b(B(M)) and b(C∗(M)) = B(M). Similarly,C(M) = b(B∗(M)) and b(C(M)) = B∗(M).

Proposition 2.0.10. Let M be a matroid on E and suppose e ∈ E.i) If e is not an isthmus, then (M − e)∗ = M∗/e.ii) If e is not a loop, then (M/e)∗ = M∗ − e.

Proposition 2.0.11. e is a loop of M if and only if e is an isthmus of M∗.

Theorem 2.0.12. Let G be a connected graph. Then the dual matroid M(G)∗ isgraphic if and only ifG is planar. Further, ifG is planar, thenM(G∗) = M(G)∗.

Definition 2.0.13. If G denotes the class of all graphic matroids, then the classof cographic matroids is defined by

G∗ = {M : M∗ ∈ G}.

Corollary 2.0.14. Let P be the class of all cycle matroids of planar graphs.Then G ∩ G∗ = P .

Theorem 2.0.15. The matrix A = [Ir×r|D] represents the matroid M preciselywhen the matrix A∗ = [−DT |I(n−r)×(n−r)] represents the dual matroid M∗.

Corollary 2.0.16. If a matroid M is representable over a field F, then M∗ isalso representable over F.

Proposition 2.0.17. The dual of a regular matroid is regular.

10

Chapter 3

Minors

Definition 3.0.1. Let M be a matroid on E with independent sets I.

Deletion: For e ∈ E (e not an isthmus), the matroid M − e has ground setE − e and independent sets that are those members of I that do not contain e:

I is independent in M − e if and only if e /∈ I and I is independent in M.

Contraction: For e ∈ E (e not a loop), the matroid M/e has ground setE − e and independent sets that are formed by choosing all those members of Ithat contains e, and then removing e from each such set:

I − e is independent in M/e if and only if e ∈ I and I is independent in M.

Definition 3.0.2. Let M be a matroid on E and T ⊂ E. The contraction M/Tis given by: M/T = (M∗\T )∗.

Corollary 3.0.3. Let BT be a basis of M |T . Then

B(M/T ) = {B′ ⊂ E − T : B′ ∪BT ∈ B(M)}

= {B′ ⊂ E − T : M |T has a basis B such that B′ ∪B ∈ B(M)}.

Proposition 3.0.4. Let BT be a basis of M |T . Then

I(M/T ) = {I ⊂ E − T : I ∪BT ∈ I(M)}

= {I ⊂ E − T : M |T has a basis B such that B ∪ I ∈ I(M)}.

Proposition 3.0.5. The circuits of M/T consist of the minimal nomempty mem-bers of {C − T : C ∈ C(M)}.

Proposition 3.0.6. For all X ⊂ E − T ,

clM/T (X) = clM(X ∪ T )− T.

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CHAPTER 3. MINORS

Proposition 3.0.7. In comparison, we have

I(M\T ) = {I ⊂ E − T : I ∈ I(M)}

C(M\T ) = {C ⊂ E − T : C ∈ C(M)}B(M\T ) is the set of maximal members of {B − T : B ∈ B(M)}

Proposition 3.0.8. For all X ⊂ E − T ,

clM\T (X) = clM(X)− T.

Proposition 3.0.9. M\T = M/T if and only if r(T ) + r(E − T ) = r(M).

Corollary 3.0.10. M\e = M/e if and only if e is a loop or a coloop of M .

Proposition 3.0.11. If e is neither an isthmus nor a loop, then M − e and M/eare both matroids.

Proposition 3.0.12. LetM be a matroid on the ground setE, with e ∈ E neitheran isthmus nor a loop.

Deletion: The bases of M − e are those bases of M that do not contain e.Contraction: The bases of M/e are those bases of M that do contain e, with

e then removed from each such basis.

Proposition 3.0.13. Let M be a matroid.i) If e is not an isthmus, then r(M − e) = r(M).ii) If e is not a loop, then r(M/e) = r(M)− 1.

Proposition 3.0.14. If T ⊂ E, then for all X ⊂ E − T ,

rM/T (X) = rM(X ∪ T )− rM(T ).

Proposition 3.0.15. Let a, b ∈ E. Assuming everything is well-defined, we have:i) (M − a)− b = (M − b)− a;ii) (M/a)/b = (M/b)/a;iii) (M/a)− b = (M − b)/a.

Proposition 3.0.16. In a matroid M , let T1 and T2 be disjoint subsets of E.Then,

i) (M\T1)\T2 = M\(T1 ∪ T2) = (M\T2)\T1;ii) (M/T1)/T2 = M/(T1 ∪ T2) = (M/T2)/T1; andiii) (M\T1)/T2 = (M/T2)\T1.

Definition 3.0.17. Any sequence of deletions and contractions from M can bewritten in the form M\X/Y for some pair of disjoint sets X and Y , either ofwhich may be empty. Matroids of this form are called minors of M .

Definition 3.0.18. If X ∪Y is nonempty, then we call M/Y \X a proper minorof M .

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CHAPTER 3. MINORS

Definition 3.0.19. A matroid N1 is called an N-minor of M if N1 is a minor ofM that is isomorphic to N .

Proposition 3.0.20. A matroid N is a minor of a matroid M if and only if N∗ isa minor of M∗. More particularly,

N = M\X/Y if and only if N∗ = M∗/X\Y.

Proposition 3.0.21. Let M be a matroid and e an element that is neither anisthmus nor a loop. Then

Circuits:a) Deletion: C is a circuit of M − e if and only if e /∈ C and C is a circuit of

M .b) Contraction: C is a circuit of M/e if and only ifi) C ∪ e is a circuit of M , orii) C is a circuit of M and C ∪ e contains no circuits except C.

Rank Function: Let A ⊂ E with e /∈ A. Thena) Deletion: rM−e(A) = rM(A).b) Contraction: rM/e(A) = rM(A ∪ e)− 1.

Deletion and Contraction in graphsDeletion: LetG−e be the graph obtained fromG by erasing the edge e. Then

the cycle matroid of G− e is the same as the matroid M(G)− e.Contraction: Let G/e be the graph from G by erasing the edge e and then

identifying its endpoints. Then the cycle matroid of G/e is the same as the ma-troid M(G)/e.

Deletion and Contraction in representable matroidsDeletion: Suppose e is not an isthmus of the matroid M [A]. Let A− e be the

matrix obtained from A by erasing the column vector corresponding to e in thematrix A. Then A− e represents the matroid M [A]− e.

Contraction: Suppose e is not a loop of M [A]. Let B be the matrix obtainedfrom A by performing elem row operations on A so that there is exactly onenon-zero entry in the column corresponding to e, and row r is the row of B con-taining the one non-zero entry of that column. Then remove both column c androw r from B. This new matrix, which we’ll call A/e, is the one we want: A/erepresents the matroid M [A]/e.

Note that e is a loop of M [A] if and only if e corresponds to the zero vector.For an isthmus, let row r of B be the row containing the one non-zero entry ofe. Then e is an isthmus of M [A] if and only if the single non-zero entry of e isalso the only non-zero entry in row r.

13

CHAPTER 3. MINORS

Definition 3.0.22. Let M be a matroid on the ground set E, and let A and B betwo disjoint subsets of E. Then M/A−B is called a minor of M .

Theorem 3.0.23. (Scum Theorem) Let N be a minor of M on the ground set E.Assume A ⊂ E is the subset of points contracted in creating the minor N . Thenthe geometric lattice for N is (isomorphic to) the upper interval [A,E] in thegeometric lattice for M .

Theorem 3.0.24. (The Scum Theorem) Let N be a minor of a matroid M . Thenthere is a subset Z of E(M)−E(N) such that M/Z and N have the same rank,and N is a restriction of M/Z. Moreover, if N has no loops, then Z can bechosen to be a flat of M .

This theorem is very useful for it enables us to view the formation of a minor Nas a two-stage process: a contraction to get the rank right followed by a deletionto remove the remaining elements not in N .

Lemma 3.0.25. Every minor of a matroidM can be written in the formM/I\I∗,where I and I∗ are independent and coindependent, respectively in M .

Proposition 3.0.26. Let T be a subset E and F be a subset of E − T . Theni) F is a flat of M/T ⇔ F ∪ T is a flat of M ;ii) F is a flat of M\T ⇔M has a flat F ′ such that F = F ′ − T .

Proposition 3.0.27. SupposeM is a matroid onE, S ⊂ E and r(M) = r(M |S).Then

M |S = M − (E − S).

Definition 3.0.28. Let X be a non-trivial parallel class of M . If we delete allthe loops from M and then, in each non-trivial parallel class X , we distinguishone element and delete all the other elements of X , the matroid we obtain isuniquely determined up to a renaming of the distinguished elements. We denotethis matroid by si(M) and call it the simplification of M . Formally, the groundset of si(M) is the set of all parallel classes of M , while a subset {X1, . . . , Xk}of these parallel classes is independent in si(M) if and only if rM(X1 ∪ X2 ∪· · · ∪Xk) = k.

Corollary 3.0.29. Suppose M is a matroid on E and let si(M) be the simplifi-cation of M . Then si(M) = M |S = M − (E − S) for some S ⊂ E.

Theorem 3.0.30. Let M be a matroid on the ground set E and let S ⊂ E. Then

M |S = (M − A)/B

where A ∪B = E − S,A ∩B = ∅, and |B| = r(M)− r(M |S).

Proposition 3.0.31. Let M and M∗ be a pair of dual matroids. Then N is aminor of M if and only if N∗ is a minor of M∗.

14

CHAPTER 3. MINORS

Definition 3.0.32. A matroid in which every subset is independent is called aBoolean algebra.

Proposition 3.0.33. A matroid M is a Boolean algebra if and only if M doesnot contain a loop as a minor.

Proposition 3.0.34. Let M = U1,1⊕U0,1 be the matroid with two points, a loopand an isthmus. Then a matroid is uniform if and only if it does not have M as aminor.

Definition 3.0.35. If a class M of matroids is closed under the operations ofdeletion and contraction, then we sayM is a hereditary class.

Definition 3.0.36. LetM be a hereditary class of matroids. Then a matroid Mis an excluded/forbidden minor for the classM if M /∈ M, but every properminor of M is inM.

Proposition 3.0.37. If G is a graph, then M(G)/T = M(G/T ) for all subsetsT of E(G).

A graph H is a minor of a graph G if H is a contraction of some subgraph of G,or equivalently, if H can be obtained from G by a sequence of edge deletions,edge contractions, and deletions of isolated vertices.

Theorem 3.0.38. A matroid M is graphic if and only if M has no minor isomor-phic to any of the following matroids:

U2,4, F7, F∗7 ,M(K5)

∗,M(K3,3)∗.

So we see that, any non-graphic matroid has one of these five matroids as a mi-nor.

Theorem 3.0.39. Graphic matroids are binary.

Theorem 3.0.40. Let M be a matroid. Then the following are equivalent.1) M is binary.2) Suppose C is a circuit and C∗ is a cocircuit. Then |C ∩ C∗| is even.3) Let C1, . . . , Ck be circuits. Then the symmetric difference

C1∆C2∆ · · ·∆Ck

is a disjoint union of circuits (possibly empty).

Corollary 3.0.41. Let C be a family of subsets of a finite set E. Then C is thecollection of circuits of a binary matroid if and only if

1) ∅ ∈ C;2) if C1, C2 ∈ C and C1 ⊂ C2, then C1 = C2;3) if C1, C2 ∈ C with C1 6= C2, then C1∆C2 is a disjoint union of elements of

C.

15

CHAPTER 3. MINORS

Corollary 3.0.42. M is binary if and only if C∗1∆C∗2 is a disjoint union of cocir-cuits for any two cocircuits C∗1 and C∗2 .

Definition 3.0.43. A subgraphH is a topological minor of a graphG if you canobtain H by deleting edges of G and also contracting edges incident to verticesof degree 2.

Theorem 3.0.44. A graph G is planar if and only if it does not contain eitherK3,3 or K5 as a topological minor.

Theorem 3.0.45. SupposeG is a graph. ThenG is planar if and only if the cyclematroid M(G) has no minor isomorphic to M(K5) or M(K3,3).

Theorem 3.0.46. Any infinite family of graphs contains a pair G and H with Ga minor of H .

Definition 3.0.47. A matrix A with integer entries is unimodular if every k× ksubmatrix has determinant 0, 1 or −1. A matroid is unimodular/regular if itcan be represented over Q by a unimodular matrix.

Theorem 3.0.48. The matroid M is regular if and only if M can be representedover every field.

Proposition 3.0.49. Let A be a matrix over a field F and T be a subset of the setE of column labels of A. We shall denote by A\T the matrix obtained from Abe deleting all the columns whose labels are in T . Then M [A]\T = M [A\T ].

Theorem 3.0.50. Graphic matroids are regular.

Theorem 3.0.51. A matroid is binary if and only if it does not contain U2,4 as aminor.

Corollary 3.0.52. A rank r matroid M is binary if and only if M has no flat ofrank r − 2 that is covered by four hyperplanes.

Corollary 3.0.53. Every minor of a graphic matroid is graphic. Every minor ofan F-representable matroid is F-representable. Every minor of a regular matroidis regular.

Definition 3.0.54. IfM\f = N and f is in a 2-circuit ofM , thenM is a parallelextension of N , and N is a parallel deletion of M . If, instead, M/f = N andf is in a 2-cocircuit of M , then M is a series extension of N , and N is a seriescontraction of M .

Definition 3.0.55. A matroid N is a series minor of M if N can be obtainedfrom M by a sequence of deletions, as well as series contractions. If N can beobtained from M by a sequence of contractions as well as parallel deletions,then N is a parallel minor of M .

Lemma 3.0.56. If N = M/y\x and {y, z} is a cocircuit of M , then N is eithera series contraction or a deletion of M\x.

16

CHAPTER 3. MINORS

Proposition 3.0.57. Let M and N be matroids. If N is a series minor of M ,then N = M\X/Y for some sets X and Y where every element of Y is in serieswith an element of M\X not in Y .

17

CHAPTER 3. MINORS

18

Chapter 4

Connectivity

Definition 4.0.1. LetM1 andM2 be matroids on disjoint ground setsE1 andE2,respectively. Define the direct sum M1 ⊗M2 to be the matroid on the groundset E = E1 ∪E2 with independent sets I1 ∪ I2, where I1 ⊂ E1 is independent inM1 and I2 ⊂ E2 is independent in M2.

Proposition 4.0.2. LetM1,M2 be matroids defined on disjoint ground setsE1, E2

with independent sets I(M1), I(M2), respectively. Then,

1) Independent sets: I(M1 ⊗M2) = {I1 ∪ I2 : Ii ∈ I(Mi)};2) Bases: B(M1 ⊗M2) = {B1 ∪B2 : Bi ∈ B(Mi)};3) Rank function: rM1⊗M2(X) = rM1(X ∩ E1) + rM2(X ∩ E2);4) Flats: F(M1 ⊗M2) = {F1 ∪ F2 : Fi ∈ F(Mi)};5) Hyperplanes: H(M1⊗M2) = {H1 ∪E2 : H1 ∈ H(M1)}∪ {E1 ∪H2 : H2 ∈H(M2)};6) Spanning sets: S(M1 ⊗M2) = {S1 ∪ S2 : Si ∈ S(Mi)};7) Circuits: C(M1 ⊗M2) = C(M1) ∪ C(M2);8) Cocircuits: C∗(M1 ⊗M2) = C∗(M1) ∪ C∗(M2);9) Duals: (M1 ⊗M2)

∗ = M∗1 ⊗M∗

2 .

Definition 4.0.3. LetM1, . . . ,Mk be matroids on disjoint ground setsE1, . . . , Ek,respectively. Define the direct sum M1 ⊕M2 ⊕ · · · ⊕Mk to be the matroid onthe ground set E = E1 ∪ · · · ∪ Ek with independent sets I1 ∪ · · · ∪ Ik, whereIj ⊂ Ej is independent in Mj for 1 ≤ j ≤ k.

Define a relation R as such: x is related to y if x = y or if there is a circuit Ccontaining both x and y.

Theorem 4.0.4. Let M be a matroid on E with circuits C and relation R givenabove. Then R is an equivalence relation on E. The R-equivalence classes arecalled the (connected) components of M .

19

CHAPTER 4. CONNECTIVITY

Definition 4.0.5. For a graph, a block is a connected graph whose cycle matroidis connected. Clearly, a loopless graph is a block if and only if it is connectedand has no cut-vertices. A block of a graph is a subgraph that is a block and ismaximal with this property.

Definition 4.0.6. A subset X of E is a separator of M if X is a union of com-ponents of M .

Proposition 4.0.7. Let T be a subset of the ground set E of a matroid M . ThenT is a separator of M if and only if

r(T ) + r(E − T ) = r(M).

Corollary 4.0.8. If T is a subset of a matroid M , then M\T = M/T if and onlyif T is a separator of M .

Lemma 4.0.9. (Circuit transitivity) Let x, y, z be distinct elements of E andC1, C2 be circuits of M with x, y ∈ C1 and y, z ∈ C2. Then there is a circuit C3

containing both x and z.

Definition 4.0.10. Let M be a matroid on the set E and let S ⊂ E. Then therestriction M |S has a ground set S, and I ⊂ S is independent in M |S if I isindependent in M .

Theorem 4.0.11. Let M be a matroid on E and suppose E = E1 ∪ · · · ∪ Ek isa partition of E, where each Ei is a circuit equivalence class under relation R.Then

M = M |E1 ⊕ · · · ⊕M |Ek,

where M |Eiis the restriction of M to the elements of Ei.

Proposition 4.0.12. Let M be a matroid on E with |E| ≥ 2. If I is an isthmusof M , then M = (M/I)⊕ I . If L is a loop of M , then M = (M − L)⊕ L.

Definition 4.0.13. A matroid M is connected if M cannot be written as a directsum of smaller matroids, i.e., if, for all x, y ∈ E, there is a circuit C containingboth x and y. If M = M1 ⊕N , where M1 is connected, then M1 is a connectedcomponent of M .

Theorem 4.0.14. Let M be a connected matroid. For all e ∈ E, either M − eor M/e (or both) is connected.

Corollary 4.0.15. A matroid M is connected if and only if its dual matroid M∗

is connected.

Theorem 4.0.16. M is a connected matroid if and only if for every pair of ele-ments x and y, there is a circuit or a cocircuit containing both.

Proposition 4.0.17. Let M be a matroid on E and suppose A ⊂ E is a propersubset satisfying r(A) + r(E − A) = r(M). Then M is disconnected, i.e.,M = M |A ⊕M |E−A.

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Proposition 4.0.18. A matroid M is disconnected if and only if for some propernonempty subset T of M , r(T ) + r∗(T )− |T | = 0.

Proposition 4.0.19. If x and y are distinct elements of a circuit C of a matroidM , then M has a cocircuit C∗ such that C ∩ C∗ = {x, y}.

Corollary 4.0.20. A matroid M is connected if and only if, for every pair ofdistinct elements of M , there is a circuit and/or a cocircuit containing both.

Proposition 4.0.21. A matroid M is disconnected if and only if E has a propernonempty subset T such that I(M) = {I1 ∪ I2 : I1 ∈ I(M |T ) and I2 ∈I(M |(E − T ))}.

Proposition 4.0.22. The classes of F-representable, graphic, cographic, transver-sal, and regular matroids are all closed under the operation of direct sum.

Theorem 4.0.23. Let G be a connected graph. Then the following are equiva-lent.

i) M(G) is a connected matroid;ii) G has no cut vertices, i.e., G is a block;iii) Every pair of vertices in G are joined by at least two vertex disjoint edge-

paths.

Proposition 4.0.24. For all matroidsM , every connected component ofM\X/Yis contained in a connected component ofM . In particular, every connected ma-troid that is a minor of M is a minor of some component of M .

Proposition 4.0.25. (M1 ⊕M2)∗ = M∗

1 ⊕M∗2 for all matroids M1 and M2.

Theorem 4.0.26. Let e be an element of a connected matroid M , and Ce be theset of circuits of M containing e. Then the circuits of M not containing e are theminimal sets of the form

(C1 ∪ C2)−⋂{C : C ∈ Ce and C ⊂ C1 ∪ C2}

where C1 and C2 are distinct members of Ce.

Corollary 4.0.27. Let N be a connected minor of a connected matroid M . Thenthere is a sequence M0,M1, . . . ,Mn of connected matroids with that M0 = Nand Mn = M such that, for all i in {0, 1, . . . , n− 1}, the matroid Mi is a singleelement deletion or a single element contraction of Mi+1.

Definition 4.0.28. A graph is connected if there is a path between every pair ofvertices. A graph G is k-connected if G remains connected after removing anyk − 1 vertices (along with all the incident edges). G is minimally k-connectedif it is k-connected, but not (k + 1)-connected.

Definition 4.0.29. A nonempty matroid M is minimally connected if M is con-nected but M\e is disconnected for all elements e.

21


Proposition 4.0.30. Let M be a minimally connected matroid. Then M has acobasis each element of which is in a 2-element cocircuit.

Lemma 4.0.31. Let M be a connected matroid having at least two elements andlet {e1, . . . , em} be a circuit of M such that M\ei is disconnected for all i in{1, 2, . . . ,m− 1}. Then {e1, . . . , em−1} contains a 2-cocircuit of M .

Theorem 4.0.32. For e in E(M), let ce(M) and c∗e(M) be the sizes of a largestcircuit and a largest cocircuit containing e. Let M be a connected matroid withat least two elements. If e is an element of M , then

|E(M)| ≤ (ce(M)− 1)(c∗e(M)− 1) + 1.

Theorem 4.0.33. In a connected matroid M with at least two elements havinglargest circuit with c elements and largest cocircuit with c∗ elements,

|E(M)| ≤ 1

2cc∗.

Proposition 4.0.34. Let G be a connected loopless graph having at least threevertices. Then G is 2-connected if and only if, for every vertex v of G, the set ofedges meeting at v is a bond.

Proposition 4.0.35. Let G be a loopless graph without isolated vertices andsuppose that |V (G)| ≥ 3. Then M(G) is a connected matroid if and only if G isa 2-connected graph.

Higher Connectivity

Definition 4.0.36. Let M be a matroid with ground set E. If X ⊂ E, let

λM(X) = r(X) + r(E −X)− r(M).

We call λM the connectivity function of M and will often abbreviate it as λ.Clearly λ(E −X) = λ(X).

Lemma 4.0.37. Let M be a matroid on E. If X ⊂ E, then

λM(X) = r(X) + r∗(X)− |X|.

Definition 4.0.38. Let k be a positive integer and M be a matroid on E. ForX ⊂ E(M), if λM(X) < k, then both X and (X,E − X) are called k-separating. Thus a 1-separating set is what we have been calling a separator.A k-separating pair (X,E −X) for which min{|X|, |E −X|} ≥ k is called ak-separation of M with sides X and E − X . Thus M is 2-connected iff it hasno 1-separations. More generally, for all integers n ≥ 2, M is n-connected iffor all k ∈ {1, . . . , n− 1}, it has no k-separations.

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Corollary 4.0.39. Let M be a matroid on E. If X ⊂ E, then

λM(X) = λM∗(X).

Moreover, M is n-connected iff M∗ is n-connected.

Recall, if X is a set of edges of a graph G and G[X], its subgraph with ω(G[X])components, then

r(X) = |V (G[X])| − ω(G[X]).

It is easy to see that an n-connected graph has no vertices of degree less than nand more generally, no bonds of size less than n.

Proposition 4.0.40. If M is an n-connected matroid and |E(M)| ≥ 2(n − 1),then all circuits and all cocircuits of M have at least n elements.

Definition 4.0.41. A k-separation (X, Y ) of a matroidM is minimal ifmin{|X|, |Y |} =k.

Corollary 4.0.42. Let (X, Y ) be a k-separation of a k-connected matroid andsuppose that |X| = k. Then X is either a coindependent circuit or an indepen-dent cocircuit.

Definition 4.0.43. For sets X and Y in a matroid M , the local connectivitybetween X and Y , denoted u(X, Y ) or uM(X, Y ), is defined by

u(X, Y ) = r(X) + r(Y )− r(X ∪ Y ).

Evidently, u(Y,X) = u(X, Y ), Moreover,

u(X,E −X) = λ(X)

and when X and Y are disjoint,

u(X, Y ) = rM(X)− rM/Y (X).

Definition 4.0.44. Sets X and Y of a matroid M are said to be skew if the rankof their union is the sum of their ranks or, equivalently, if u(X, Y ) = 0.

Lemma 4.0.45. Let X1, X2, Y1, and Y2 be subsets of the ground set of a matroidM . If X1 ⊃ Y1 and X2 ⊃ Y2, then

u(X1, X2) ≥ u(Y1, Y2)

or equivalently,

r(X1) + r(X2)− r(X1 ∪X2) ≥ r(Y1) + r(Y2)− r(Y1 ∪ Y2).

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Corollary 4.0.46. Let z be an element of a matroid M and let (X, Y ) be apartition of E(M)− z into possibly empty sets. Then

λM(X)− 1 ≤ λM\z(X) ≤ λM(X)

andλM(X)− 1 ≤ λM/z(X) ≤ λM(X).

Moreover, in each of i)-iv), parts a)-c) are equivalent:i) a) λM\z(X) = λM(X);b) z ∈ clM(Y ) or z is a coloop of M ;c) z 6∈ cl∗M(X) or z is a coloop of M ;

ii) a) λM\z(X) = λM(X)− 1;b) z 6∈ clM(Y ) and z is not a coloop of M ;c) z ∈ cl∗M(X) and z is not a coloop of M ;

iii) a) λM/z(X) = λM(X);b) z ∈ cl∗M(Y ) or z is a loop of M ;c) z 6∈ clM(X) or z is a loop of M ;

iv) a) λM/z(X) = λM(X)− 1;b) z 6∈ cl∗M(Y ) and z is not a loop of M ;c) z ∈ clM(X) and z is not a loop of M .

The only ways to destroy n-connectedness by single-element extensions are byadding a coloop or creating a small circuit.

Proposition 4.0.47. Let e be an element of a matroid M . Suppose that M\e isn-connected but M is not. Then either e is a coloop of M , or M has a circuitthat contains e and has fewer than n elements.

Proposition 4.0.48. If e is an element of an n-connected matroid M , then pro-vided |E(M)| ≥ 2(n− 1), both M\e and M/e are (n− 1)-connected.

Lemma 4.0.49. If X and Y are subsets of E, then

λ(X) + λ(Y ) ≥ λ(X ∪ Y ) + λ(X ∩ Y ).

Corollary 4.0.50. Let X and Y be k-separating sets. If one of X∪Y and X∩Yis not (k − 1)-separating, then the other is k-separating.

Definition 4.0.51. In a matroid M , a set X or a pair (X,E − X) is exactlyk-separating if λM(X) = k − 1. Similarly, a k-separation (X, Y ) is exact ifλM(X) = k − 1.

24


Proposition 4.0.52. Let z be an element of a matroid M and let (X, Y ) be apartition of E(M)− z into possibly empty sets. Then

λM(X)− 1 ≤ λM(X ∪ z) ≤ λM(X) + 1.

Moreover, in each of i)-iii), parts a)-c) are equivalent:i) a) λM(X ∪ z) = λM(X);b) z is in exactly one of clM(X) and cl∗M(X);c) z is in exactly one of clM(X) ∩ clM(Y ) and cl∗M(X) ∩ cl∗M(Y ).

ii) a) λM(X ∪ z) = λM(X)− 1;b) z ∈ clM(X) ∩ cl∗M(X);c) z 6∈ clM(Y ) ∪ cl∗M(Y ).

iii) a) λM(X ∪ z) = λM(X) + 1;b) z 6∈ clM(X) ∪ cl∗M(X);c) z ∈ clM(Y ) ∩ cl∗M(Y ).

Theorem 4.0.53. Let M and N be connected matroids with the same ground setand the same connectivity function. If 1) M is binary, or ii) r(M) 6= r∗(M),then M = N , or M = N∗.

Definition 4.0.54. LetM andN be matroids each with at least two elements. LetE(M)∩E(N) = {p} and suppose that neitherM norN has {p} as a separator.Then the 2-sum M ⊕2 N is the matroid with ground set (E(M) ∪ E(N)) − pand set of circuits

C(M\p) ∪ C(N\p) ∪ {(C ∪D)− p : p ∈ C ∈ C(M) and p ∈ D ∈ C(N)}.

Theorem 4.0.55. A 2-connected matroid M is not 3-connected if and only ifM = M1⊕2M2 for some matroids M1 and M2, each of which has at least threeelements and is isomorphic to a proper minor of M .

Lemma 4.0.56. Let C1 and C2 be circuits of M each of which meets both X1

and X2. Then C1 ∩X1 is not a proper subset of C2 ∩X1.

Lemma 4.0.57. Let Y1 and Y2 be non-empty subsets of X1 and X2, respectively.Suppose that M has circuits C1 and C2 with C1 ∩X1 = Y1 and C2 ∩X2 = Y2such that C1 ∩X2 and C2 ∩X1 are non-empty. Then Y1 ∪ Y2 is a circuit of M .

Corollary 4.0.58. Every matroid that is not 3-connected can be constructed from3-connected proper minors of itself by a sequence of the operations of direct sumand 2-sum.

Proposition 4.0.59. Let M,N,M1 and M2 be matroids such that M = M1 ⊕2

M2 and N is 3-connected. If M has an N -minor, then M1 or M2 has an N -minor.

25


Corollary 4.0.60. Let M1 ⊕2 N1 = M2 ⊕2 N2 where E(Mi) ∩ E(Ni) = {pi}for each i. If E(M1)− p1 = E(M2)− p2, then M2 and N2 can be obtained fromM1 and N1 by relabelling p1 by p2 in each.

Proposition 4.0.61. Let M ′ be a matroid that is obtained by relaxing a circuit-hyperplane of a matroid M . If M is n-connected, then so is M ′.

Definition 4.0.62. A triangle is a 3-element circuit, and a triad is a 3-elementcocircuit.

Definition 4.0.63. For disjoint subsets X and Y of E of a matroid M , let

κM(X, Y ) = min{λM(S) : X ⊂ S ⊂ E − Y }

If S is a set for which this minimum is attained where X ⊂ S ⊂ E − Y , thenclearly

κM(X, Y ) = λM(S) = κM(S,E − S).

In general, we have

κM(Y,X) = κM(X, Y ) and κM∗(X, Y ) = κM(X, Y ).

Theorem 4.0.64. (Tutte’s Linking Theorem) Let X and Y be disjoint subsets ofthe ground set of a matroid M . Then κM(X, Y ) equals the maximum value ofκN(X, Y ) over all minors N of M with E(N) = X ∪ Y .

Lemma 4.0.65. Let e be an element of a matroid M , and X and Y be subsets ofE(M)− e. Then

λM\e(X) + λM/e(Y ) ≥ λM(X ∩ Y ) + λM(X ∪ Y ∪ e)− 1.

Lemma 4.0.66. Let X ′ and Y ′ be disjoint subsets of E. If X ⊂ X ′ and Y ⊂ Y ′,then κM(X, Y ) ≤ κM(X ′, Y ′).

Lemma 4.0.67. Let N be a minor of a matroid M , and let X ′ and Y ′ be disjointsubsets of E(M). If X and Y are subsets of E(N) with X ⊂ X ′ and Y ⊂ Y ′,then

κN(X, Y ) ≤ κM(X ′, Y ′).

Theorem 4.0.68. LetX and Y be disjoint sets of elements in a matroidM . ThenM has a minorN with ground setX∪Y for which κN(X, Y ) = κM(X, Y ) suchthat N |X = M |X and N |Y = M |Y .

Lemma 4.0.69. The following are equivalent for a matroid M .i) κ(M) = r(M).ii) M has no two disjoint cocircuits.iii) Every cocircuit of M is spanning.iv) M cannot be written as the union of two proper flats.

26


Definition 4.0.70. If a matroid M has a k-separation for some k, we define theTutte connectivity τ(M) ofM to bemin{j : M has a j-separation}; otherwisewe take τ(M) to be∞.

M is n-connected iff τ(M) ≥ n, also τ(M∗) = τ(M).

Definition 4.0.71. The girth g(M) of a matroid M is the minimum circuit sizeof M unless it has no circuits, in which case g(M) =∞.

Theorem 4.0.72. Let M be a matroid and suppose that M is not isomorphic toany uniform matroid Ur,n with n ≥ 2r − 1. Then

τ(M) = min{κ(M), g(M)}.

Definition 4.0.73. Let M be a 3-connected matroid. We call an element e of Messential if neither M\e nor M/e is 3-connected.

Lemma 4.0.74. (Bixby’s Lemma) Let e be an element of a 3-connected matroidM . Then either M\e or M/e has no non-minimal 2-separations. Moreover,in the first case, co(M\e) is 3-connected, while, in the second case si(M/e) is3-connected.

Corollary 4.0.75. If e is an element of a 3-connected matroid M and neitherM\e nor M/e is 3-connected, then e is in a triangle or a triad of M .

Lemma 4.0.76. A 3-connected matroid M contains a set that is both a triangleand a triad if and only if M ∼= U2,4.

Lemma 4.0.77. (Tutte’s Triangle Lemma) Let M be a 3-connected matroid hav-ing at least four elements and suppose that {e, f, g} is a triangle of M such thatneither M\e nor M\f is 3-connected. Then M has a triad that contains e andexactly one of f and g.

Theorem 4.0.78. (Tutte’s Wheels-and-Whirls Theorem) The following are equiv-alent for a 3-connected matroid M having at least one element.

i) For every element e of M , neither M\e nor M/e is 3-connected.ii) M has rank at least three and is isomorphic to a wheel or a whirl.

Definition 4.0.79. In a simple, cosimple matroidM , a subset S of E(M) havingat least three elements is a fan in M if there is an ordering (s1, . . . , sn) of theelements of S such that, for all i ∈ {1, 2, . . . , n− 2},

i) {si, si+1, si+2} is a triangle or a triad; andii) when {si, si+1, si+2} is a triangle, {si+1, si+2, si+3} is a triad; and, when

{si, si+1, si+2} is a triad, {si+1, si+2, si+3} is a triangle.The ordering (s1, s2, . . . , sn) is called a fan ordering of S, or a fan.

27


28

Chapter 5

Graphic Matroids

Proposition 5.0.1. If G is a graph, then M(G) is representable over every field.

Let AD(G) denote the incidence matrix of the directed graph D(G), that is, aij is1 if vertex i is the tail of a nonloop arc j; −1 if vertex i is the head of nonlooparc j; 0 otherwise.

Proposition 5.0.2. Let D(G) be an arbitrary orientation of a graph G, and letF be a field. Then AD(G) represents M(G) over F.

Proposition 5.0.3. If G is a graph, then M(G) is regular and M∗(G) is regular.

Proposition 5.0.4. Let G∗ be a geometric dual of a planar graph G. Then

M(G∗) ∼= M∗(G).

Theorem 5.0.5. A graph G is planar if and only if M∗(G) is graphic.

Theorem 5.0.6. A graph is planar if and only if it has no minor isomorphic toK5 or K3,3.

Theorem 5.0.7. The following are equivalent for a graph G:i) G is a planar graph;ii) M(G) is a planar-graphic matroid;iii) M(G) has no minor isomorphic to M(K5) or M(K3,3).

Corollary 5.0.8. The class of planar-graphic matroids is minore-closed.

Theorem 5.0.9. Let G be a graph with edge set E, and let I be the collection ofall subsets of E that do not contain a cycle. Then I forms the independent setsof a matroid on the ground set E, called the cycle matroid M(G).

Theorem 5.0.10. If T is a tree with n vertices, then T has n− 1 edges.

29

CHAPTER 5. GRAPHIC MATROIDS

Proposition 5.0.11. Let G be a graph with cycle matroid M(G). Then there isa connected graph G′ with M(G) ∼= M(G′).

Proposition 5.0.12. Let G be a graph and let e be an edge that is neither a loopnor an isthmus in the cycle matroid M(G). Then

M(G)− e = M(G− e) and M(G)/e = M(G/e).

Theorem 5.0.13. Let C1 and C2 be circuits in the cycle matroid M(G) of agraph G. Then C1∆C2 is a disjoint union of circuits.

Proposition 5.0.14. Let B be a basis of a matroid M = (E,B).i) For all x /∈ B, there is a unique circuit C ⊂ B ∪ x which contains x.ii) For all y ∈ B there is a unique cocircuit C∗ ⊂ E − (B − y).

Proposition 5.0.15. For every circuit C and cocircuit C∗ of the matroid M ,|C ∩ C∗| 6= 1.

Proposition 5.0.16. Let B1, B2 be bases of the matroid M = (E,B) with x ∈B1 −B2 and y ∈ B2 −B1. The following are equivalent:

i) B1 − x ∪ {y} is a basis;ii) x is in the basic circuit C of y with respect to B1;iii) y is contained in the cocircuit C∗ = E −B1 − x.

Proposition 5.0.17. LetM(G) be a graphic matroid with circuitC and cocircuitC∗. Then |C ∩ C∗| is even.

Theorem 5.0.18. Let G be a connected graph. Then the dual matroid M(G)∗ isgraphic if and only ifG is planar. Further, ifG is planar, thenM(G∗) = M(G)∗.

Theorem 5.0.19. Let G be a graph with not cut-vertices.i) The symmetric difference of any two minimal cut-sets is a disjoint union of

minimal cut-sets.ii) Every minimal cut-set is a symmetric difference of vertex cut-sets i.e., ev-

ery cocircuit of the cycle matroid M(G) is a symmetric difference of the vertexcocircuits.

iii) Let S ⊂ V and let C∗S be the symmetric difference of the vertex cut-setsover the vertices of S. Then C∗S = C∗V−S .

Definition 5.0.20. Let G be a graph. The cocircuit matroid M∗(G) is the ma-troid whose circuits are the cocircuits of the cycle matroid M(G).

Theorem 5.0.21. Let G be a graph with cycle matroid M(G). Then M∗(G) =M(G)∗, i.e., the cocircuit matroid associated toG is the dual of the cycle matroidM(G).

Definition 5.0.22. If G has m vertices and n edges, the vertex-edge incidencematrix AG is an m× n matrix constructed as follows:

i) Label the rows by the m vertices and the columns by the n edges.ii) Set ai,j = 1 if vertex i is incident to edge j, and set ai,j = 0 otherwise.

30


For a graph G having no isolated vertices, we have the following operations onG:

Vertex identification: Let v and v′ be vertices of distinct components of G.We modify G by identifying v and v′ as a new vertex v.

Vertex cleaving: This is the reverse operation of vertex identification, so agraph can only be cleft at a cut-vertex or at a vertex incident with a loop.

Twistin: Suppose that the graph G is obtained from disjoint graphs G1, G2

by identifying the vertices u1 of G1 and u2 of G2 as the vertex u of G, and iden-tifying the vertices v1, v2 as the vertex v of G. In a twisting of G about {u, v},we identify, instead u1 with v2, and v1 with u2. We call G1, G2 pieces of thetwisting.

Definition 5.0.23. Two graphs G and G′ are 2-isomorphic if G can be trans-formed toG′ by a sequence of the three operations of vertex identification, vertexsplitting and twisting.

Proposition 5.0.24. Suppose G is 2-isomorphic to G′. Then M(G) ∼= M(G′).

Theorem 5.0.25. (Whitney’s 2-Isomorphism Theorem) Let G and H be graphshaving no isolated vertices. Then M(G) and M(H) are isomorphic if and onlyif G and H are 2-isomorphic.

Theorem 5.0.26. Suppose G is a 3-connected graph with no loops. Then G canbe uniquely reconstructed from its cycle matroid M(G).

31


32

Chapter 6

Finite Geometry

Definition 6.0.1. Let q = pk for some prime p and positive integer k. The pointsof the affine planeAG(2, q) are the q2 ordered pairs (x, y), where x, y ∈ Fq. Thelines are collections of points (x, y) satisfying equations of the form ax+by = c,where a, b, c ∈ Fq.

Definition 6.0.2. Let F ⊂ Fnq . Then F is a flat if F = u+W = {u+w : w ∈ W}

for some subspace W ⊂ Fnq and some vector u ∈ Fn

q . We also call F = ∅ a flat.

So in Rn, the one-dimensional subspaces are lines through the origin, thetwo-dimensional subspaces are planes through the origin, and so on. Then ourflats will just be translates of these subspaces. In general, the flats are cosets ofthe quotient group V/W .

Theorem 6.0.3. Let M be defined on the ground set E, where E consists of theqn points of AG(n, q). Let F be the family of all flats of Fn

q along with ∅. ThenF are the flats of a matroid.

Flat-matrix Correspondence:

Given a flat F inAG(n, q), find a vector u and a subspaceW so that F = u+W .Next, suppose dim(W ) = r. Then dim(W⊥) = n− r, where W⊥ = {v ∈ Fn

q :w · v = 0 for all w ∈ W}.Now find a basis for W⊥ and form an (n−r)×n matrix A whose rows are thesebasis vectors for W⊥.Then the null space N(A) = W (since (W⊥)⊥ = W ). This means W = {x ∈Fnq : Ax = 0}.

Now add u: this gives F = {y ∈ Fnq : y = x + u and Ax = 0}. Equivalently,

F = {y ∈ Fnq : Ay = Au}.

33

CHAPTER 6. FINITE GEOMETRY

Proposition 6.0.4. For every rank r flat F in AG(n, q), there is an (n− r)× nmatrix A and a vector u so that F is the set of all n-tuples y ∈ Fn

q satisfyingAy = Au.

Proposition 6.0.5. Let S = {P0, . . . , Pk} be a collection of points in the affinegeometry AG(n, q). Then S is affinely dependent if and only if there are scalarsa0, . . . , ak ∈ Fq, not all 0, such that both of the following conditions hold:

1)∑k

i=0 ai = 0, and2)∑k

i=0 aiPi = 0

Proposition 6.0.6. A set {P0, . . . , Pk} of k+1 points is affinely dependent if andonly if the corresponding k vectors {v1, . . . , vk} are linearly dependent, wherevi = Pi − P0.

Proposition 6.0.7. Let x = (x1, . . . , xn) ∈ Fnq be an n-tuple, and define an

(n + 1)-tuple x′ ∈ Fn+1q by placing a 1 in the first coordinate of x : x′ =

(1, x1, . . . , xn). Then the vectors v1, . . . , vm are affinely dependent in Fnq if and

only if the corresponding vectors v′1, . . . , v′m are linearly dependent in Fn+1

q .

Proposition 6.0.8. Suppose thatE is a set that labels a multiset of elements fromV (m,F). Let I be the collection of subsetsX ofE such thatX labels an affinelyindependent subset of V (m,F). Then (E, I) is a matroid.

Definition 6.0.9. The matroid (E, I) above, is called the affine matroid on E.If M is isomorphic to such a matroid, we say that M is affine over F.

In general, if M is an affine matroid over R of rank m+ 1 where m ≤ 3, then asubset X ⊂ E is dependent in M if, in the representation of X by points in Rm,there are two identical points, or three collinear points, or four coplanar points,or five points in space.

Proposition 6.0.10. Let S = {P1, . . . , Pn} be a collection of points in the affinegeometry AG(n, q).

i) Rank: r(S) = dim(F ) + 1, where F is the smallest flat containing S.ii) Closure: P ∈ S if and only if there are scalars ai ∈ Fq witha)∑n

i=1 ai = 1, andb) P =

∑ni=1 aiPi.

iii) Hyperplanes: S is a hyperplane if and only if S is the set of all n-tuples(x1, . . . , xn) ∈ Fn

q satisfying a1x1 + a2x2 + · · · + anxn = b for some scalarsa1, . . . , an, b ∈ F.

Definition 6.0.11. P ∈ Fnq is an affine combination of S = {P1, . . . , Pn} if∑n

i=1 ai = 1 and P =∑n

i=1 aiPi, i.e., P ∈ S.

34


Proposition 6.0.12. Let x = (x1, . . . , xn) ∈ Fnq be an n-tuple, and, as before,

define an (n+ 1)-tuple x′ ∈ Fn+1q by placing a 1 in the first coordinate of x. Let

u, v1, . . . , vm ∈ Fnq . Then u is an affine combination of v1, . . . , vm if and only if

the corresponding vector u′ is a linear combination of v′1, . . . , v′m in Fn+1

q .

Definition 6.0.13. A projective plane is a triple (P ,L, I) of points P , lines Land incidence I between points and lines satisfying

P1) Every pair of points determines a unique line, andP2) Every pair of lines intersect in a unique point,

and contains a four-point, i.e., there are at least four points, no three of whichare on a line.

Note that there are no pairs of parllel lines, since every pair intersects. Also,every line has the same number of points. If each line has q + 1 points, we saythe projective plane has order q.

Definition 6.0.14. Let q = pk for some prime p. The points of the projectiveplane PG(2, q) are the one-dimensional subspaces of the three-dimensional vec-tor space F3

q , and the lines are the two-dimensional subspaces.

The points of PG(2, q) (which we can think of as vectors in F3q) form the

ground setE of a matroid and a collection of points is independent in the matroidif and only if the corresponding vectors in F3

q are linearly independent.

Proposition 6.0.15. Let PG(2, q) be the projective plane over the field Fq.1) PG(2, q) is a matroid whose ground set is the set of points of the geometry.2) Every pair of points determines a unique line, and every pair of lines in-

tersects in exactly one point.3) PG(2, q) has q2 + q + 1 points and q2 + q + 1 lines.4) Three points are collinear if and only if the corresponding vectors are

linearly dependent.5) For scalars a, b, c ∈ Fq, not all zero, the equations ax+by+cz = 0 defines

a line.6) Removing any line from PG(2, q) leaves the affine plane AG(2, q).

Definition 6.0.16. The projective geometry PG(n, q) can be defined as fol-lows:

i) Points: the points of PG(n, q) are the lines through the origin in theaffine space AG(n + 1, q) (i.e., the one-dimensional subspaces of teh (n + 1)-dimensional vector space Fn+1

q ).ii) Lines: the lines of PG(n, q) are the (two-dimensional) planes through the

origin in the affine space AG(n+ 1, q).iii) k-dimensional planes: the k-dimensional planes of PG(n, q) are the (k+

1)-dimensional planes through the origin in the affine space AG(n+ 1, q).

35


Definition 6.0.17. Let V be a vector space over a field F. The projective ge-ometry PG(V ) associated with V consists of a set of points, a disjoint set oflines, and an incidence relation between them. The construction of PG(V ) fromV is analogous to the construction of si(M) from a matroid M . If V is finiteand we view both V and PG(V ) as matroids, these constructions are identical.An alternative way to construct PG(V ) from V is to first delete the zero vectorand then, from each 1-dimensional subspace, delete all but one of the remainingelements. The elements that are left become the points of the projective geometry.

Definition 6.0.18. A projective geometry/space is a triple (P,L, i) where Pand L are disjoint sets of points and lines, and i an incidence relations in whichthe following holds:

i) Every two distinct points, a and b, are on exactly one line ab.ii) Every line contains at least three points.iii) If a, b, c and d are four distinct points, no three of which are collinear, and

if the line ab intersects the line cd, then the line ac intersects the line bd.

Definition 6.0.19. A subspace of a projective geometry (P,L, i) is a subset P1

of P such that if a and b are distinct elements of P1, then all points on the lineab are in P1.

Definition 6.0.20. The subspaces of (P,L, i), ordered by inclusion, form a par-tially ordered set having a zero. The (projective) dimension of a subspace is oneless than its height in this poset provided the height is finite. If V = V (n+ 1,F),then PG(V ) has dimension n and is denoted PG(n,F).

Theorem 6.0.21. Every finite projective geometry of dimension greater than twois isomorphic to PG(n, q) for some integer n exceeding two and some primepower q.

For every finite subset S of PG(m−1,F), there is a matroid induced on S bylinear independence over F. We shall denote this matroid by PG(m− 1),F)|S.

Theorem 6.0.22. Let M be a simple rank r matroid and F be a field. The fol-lowing are equivalent:

i) M is F-representable.ii) PG(r − 1,F) has a finite subset T such that M ∼= PG(r − 1,F)|T .iii) For some m ≥ r, there is a finite subset S of PG(m − 1,F) such that

M ∼= PG(m− 1,F)|S.

In view of this result, we see that a study of F-representable simple matroidsis a study of the restrictions of projective geometries over F. Just as every n-vertex simple graph can be obtained from Kn by deleting edges, so too canevery rank r simple F-representable matroid be obtained from PG(r − 1,F) bydeleting elements. Also, most of the time, we will view PG(r − 1, q) as thesimple matroid associated with V (r, q).

The number of points in PG(n, q) is qn+1−1q−1 = qn + qn−1 + · · ·+ q + 1.

36


Definition 6.0.23. Let n be an integer exceeding −2. The affine geometryAG(n,F) is obtained from PG(n,F) by deleting all the points of a hyperplane ofthe latter. Equivalently, we can construct AG(n,F) directly from V (n+ 1,F) asfollows: Delete a hyperplaneH of V (n+1,F), then for each 1-dimensional sub-space X that is not contained in H , choose a single representative from X −H .These representatives are the points of AG(n,F).

In particular, as V (n + 1,F) = {(x1, x2, . . . , xn+1) : xi ∈ F}, the set ofvectors (x1, . . . , xn+1) with x1 = 0 is a hyperplane. With this deleted, all re-maining vectors have non-zero first coordinant. If we pick, as the represen-tative for each 1-dimensional subspace, that vector whose first non-zero coor-dinate is a one, then we see that the point set of AG(n,F) can be viewed as{(1, x2, x3, . . . , xn+1) : xi ∈ F}. Now let S be a finite subset of this. Then thereis a matroid, AG(n,F)|S, induced on the points of S by linear dependence overF. This is precisely an affine matroid. A matroid M is affine over F if and onlyif M has no loops and si(M) ∼= AG(n,F)|S for some n ≥ −1 and some finitesubset S of AG(n,F).

Corollary 6.0.24. A simple rank r matroid M that is representable over GF (q)has at most qr−1

q−1 elements. Moreover, if |E(M)| = qr−1q−1 , then M is isomorphic

to PG(r − 1, q).

Corollary 6.0.25. If r ≥ 2 and e ∈ PG(r − 1, q), then PG(r − 1, q)/e isthe matroid that is obtained from PG(r − 2, q) by replacing each element by qelements in parallel.

Proposition 6.0.26. Let E be the set of points of PG(n, q), and let F be thecollection of all planes of all dimensions of PG(n, q), along with ∅ and E. ThenF is the family of flats of a matroid.

Proposition 6.0.27. Let H be a hyperplane of PG(n, q).1) H is isomorphic to PG(n− 1, q).ii) PG(n, q)−H is isomorphic to AG(n, q).

Theorem 6.0.28. The lattice of subspaces of Fn+1q ordered by inclusion is a

geometric lattice, i.e., the lattice of flats of the matroid PG(n, q).

Proposition 6.0.29. Let F be a flat in PG(n, q) with r(F ) = k. Then there isan (n− k)× n matrix A such that F = {v : Av = 0}.Definition 6.0.30. Denote by

[nk

]q

the number of k-dimensional subspaces of then-dimensional vector space Fn

q . These numbers are called q-binomial coeffi-cients.

Theorem 6.0.31. The number of k-dimensional subspaces of the n-dimensionalvector space Fn

q is[n

k

]q

=(qn − 1)(qn−1 − 1) · · · (qn−k+1 − 1)

(qk − 1)(qk−1 − 1) · · · (q − 1).

37


Theorem 6.0.32. The number of flats of rank k in PG(n, q) is[n+1k

]q.

Theorem 6.0.33. Let 1 ≤ k ≤ n. Then the number of flats of rank k inAG(n, q)is qn−k+1

[n

k−1

]q.

Proposition 6.0.34. For all non-negative integers k, n, with k ≤ n, we have[nk

]q

=[

nn−k

]q.

Proposition 6.0.35.[n+1k

]q

=[

nk−1

]q

+ qk[nk

]q.

Theorem 6.0.36. Let P be a projective plane, and assume some line of P con-tains exactly m+ 1 points. Then

1) Every line has m+ 1 points.2) Every point is on exactly m+ 1 lines.3) There are a total of m2 +m+ 1 points.4) There are a total of m2 +m+ 1 lines.

We say P is a projective plane of order m.

Theorem 6.0.37. A projective plane of order n exists if and only if an affineplane of order n exists.

Theorem 6.0.38. (Pappus’ Theorem) Let a, b, c, d, e, f be points in a projectiveplane PG(2, q) such that a, b, c and d, e, f are collinear. Let g = ae ∩ bd, h =af ∩ cd and i = bf ∩ ce. Then g, h and i are collinear.

Theorem 6.0.39. (Desargues’ Theorem) Let bg, eh, and fi be three lines inPG(2, q) all meeting at the point a. Let j = bh∩eg, d = fh∩ei and c = bf∩gi.Then c, d and j are collinear.

Theorem 6.0.40. Suppose P is a finite projective plane. Then the following areequivalent:

1) P = PG(2, q) for some prime power q.2) Desargues’ Theorem holds.3) Pappus’ Theorem holds.

Theorem 6.0.41. (Veblen-Young Theorem) Every finite projective space of di-mension at least three is isomorphic to the projective space PG(n, q) for somen ≥ 3 and q a prime power.

38

Chapter 7

Representable Matroids

Proposition 7.0.1. Let A be an r×n matrix, and let A′ be the reduced row ech-elon form of the matrix A. Then a subset of columns of A is linearly independentif and only if the corresponding columns of A′ are.

Definition 7.0.2. In general, a rank-r matroid M is F-representable if and onlyif for somem ≥ r, there is a function ψ : E(M)→ V (m,F) such that rM(X) =dim〈ψ(X)〉 for all X ⊂ E. Such a map ψ is called a coordinatization of Mover F. Clearly if φ is a coordinatization ofM , then r = dim(φ(E(M))). Hencethe rank r matroid is F-representable if and only if M has a coordinatization φover F such that φ : E(M)→ V (r,F).

A matroid is F-representable if and only if its associated simple matroidsi(M) is F-representable. Hence, we usually concentrate on simple matroids.Now suppose that the matroid M is simple and that φ : E(M) → V (m,F) is acoordinatization of M over F. Then φ is one-to-one. Moreover, φ(E(M)) doesnot contain 0, nor does it contain more than one element of any 1-dimensionalsubspace of V (m,F). Thus, when F is finite, we can view φ(E(M)) as being arestriction of the simple matroid associated with V (m,F). This simple matroidis the matroid associated with the projective geometry PG(m− 1,F).

For a non-zero matrix A over a field F, we have seven operations on A thatdo not alter M [A]. Six of these are standard matrix operations:

i) Interchange two rows.ii) Multiply a row by a non-zero member of F.iii) Replace a row by the sum of that row and another.iv) Adjoin or remove a zero row.v) Interchange two columns (moving their labels along).vi) Multiply a column by a non-zero member of F.

39

CHAPTER 7. REPRESENTABLE MATROIDS

vii) Replace each matrix entry by its image under some automorphism of F.

Definition 7.0.3. Let M be a rank r matroid on the set {e1, . . . , en} wherer ≥ 1. For each i in {1, 2}, let Ai be an ri × n matrix over the field F whosecolumns are labelled,in some order, by e1, . . . , en. Assume that the identity mapon {e1, . . . , en} is an isomorphism between M and M [Ai] for each i. We defineA1 andA2 to be equivalent representations ofM ifA2 can be obtained fromA1

by a sequence of the operations i-vii. Thw two representations are projectivelyequivalent if A2 can be obtained from A1 by a sequence of these operations thatdoes not include vii.

Proposition 7.0.4. Suppose that r ≥ 1 and A1 and A2 are r × n matrices overa field F with the columns of each matrix being labelled, in order, by e1, . . . , en.Then A1 and A2 are projectively equivalent representations of a matroid on{e1, . . . , en} if and only if there are a non-singular r × r matrix X and a non-singular n× n diagonal matrix Y such that A2 = XA1Y .

A representation has a very useful geometric interpretation. Let M be an F-representable matroid of rank r ≥ 3. Roughly, two representations of M areequivalent if one is the image of the other under an automorphism of PG(r −1,F) where an automorphism of this PG is a permutation of its set of pointsthat maps lines onto lines. These automorphisms are often called collineations.

Definition 7.0.5. A permutation σ of V (r,F) is a non-singular semilinear trans-formation if there are a non-singular linear transformation τ of V (r,F) and anautomorphism α of F such that if v ∈ V (r,F) and τ(v) = (w1, . . . , wr), thenσ(v) = (α(w1), . . . , α(wr)).

Proposition 7.0.6. Let M be a matroid on the set {e1, . . . , en}. Suppose thatr ≥ 1 and let A1 and A2 be r × n matrices over a field F such that, for eachi on {1, 2}, the map taking ej to the jth column v(i)j of Ai is an isomorphismfrom M to M [Ai]. Then A1 and A2 are equivalent representations of M if andonly if there are a non-singular semilinear transformation σ of V (r,F) and asequence c1, . . . , cn of non-zero elements of F such that v(2)j = cjσ(v

(1)j ) for all

j in {1, 2, . . . , n}.

Theorem 7.0.7. If r ≥ 3, then every automorphism of PG(r − 1,F) is inducedby a non-singular semilinear transformation of V (r,F).

Definition 7.0.8. A rank r matroid M on an n-element set is called uniquely F-representable if all of the r×n matrices representing M over F are equivalent.

For A a matrix, and e not an isthmus, we define A− e to be the matrix obtained

40


by removing e from A. For contraction, suppose e is not a loop. A/e is definedas follows: first, row reduce A so that there is exactly one non-zero entry in thecolumn e and this non-zero entry appears in the first row. Further, assume thisnon-zero entry equals 1. Then remove both column and row (from the row re-duced matrix). This is now A/e.

Proposition 7.0.9. Let A be a matrix and let e be a column vector of A that isneither a loop nor an isthmus in the matroid M [A]. Defining the matrices A− eand A/e as above, we have

M [A]− e = M [A− e] M [A]/e = M [A/e].

When e is a loop, e must correspond to a column of 0’s. The column e is anisthmus in M [A] if, when row reducing A so that e has exactly one non-zeroentry, that entry is also the only non-zero entry in its row.

Theorem 7.0.10. The matrix A = [Ir×r|D] represents the matroid M preciselywhen the matrix A∗ = [−DT |I(n−r)×(n−r)] represents the dual matroid M∗.

Lemma 7.0.11. Let A be an r × n matrix of rank r, and let M = M [A]. Let{v1, . . . , vk} be a basis for the null space N(A), and let A∗ be the matrix withrows v1, . . . , vk. Then M [A∗] = M [A]∗, i.e., a subset S of columns in A∗ is abasis for M [A]∗ if and only if the complementary columns E − S of A are abasis for M [A].

Proposition 7.0.12. Let A be an r × n matrix over a field F, let B be an r × rinvertible matrix and let D be an n× n invertible diagonal matrix.

1) The column dependences of A are the same as those of BA.2) The column dependences of A are the same as those of AD.

Theorem 7.0.13. Let M be a rank r matroid on n points representable over afield F. Then M is represented by an r× n matrix A = [I|A′], and the entries inA′ that correspond to a coordinatizing path can all be taken to be 1.

Coordinatizing Path Algorithm:

For a matroid M on elements a, . . . , h:1) First, pick a basis for M , say abc.2) Next, find the basic circuits that contain each of the remaining points.3) Now record the incidence between the points d, e, . . . , h and their basic

circuits in two ways: via a matrix A′ and a bipartite graph G.4) Find a spanning tree for the bipartite graph G, if possible.5) Highlight the entries of the matrix A′ that correspond to the edges of the

spanning tree. Replace each of those entries by a 1.

41


6) You can now assume that your representing matrix A has the form [I|A′].

If B is the basis {e1, . . . , er} of M , then it is natural to have a matrix [Ir|D].If we let D# be the matrix obtained from D by replacing each non-zero entryof D by a 1, then the columns of D# are precisely the incidence vectors of thesets C(ek, B) − ek where the rows and columns of D# inherit their labels fromD. D# is called the B-fundamental-circuit incidence matrix of M . [Ir|D#] issometimes called a partial representation for M .

Proposition 7.0.14. Let M be a matroid on a set E and B be a basis of M . IfX is the B-fundamental circuit incidence matrix of M , then XT is the (E−B)-fundamental circuit incidence matrix of M∗.

Let G(D#) denote the associated simple bipartite graph, that is, it has vertexclasses {e1, . . . , er} and {er+1, . . . , en}, and two vertices ei and ej are adjacentif and only if the entry in row ei and column ej of D# is 1.

Theorem 7.0.15. For a field F, let the r×nmatrix [Ir|D1] be an F-representationof M . Let {b1, . . . , bk} be a basis of the cycle matroid of G(D#

1 ). Then k = n−ω(G(D#

1 )). Moreover, if (θ1, . . . , θk) is an ordered k-tuple of non-zero elementsof F, thenM has a unique F-rep [Ir|D2] that is projectively equiv to [Ir|D1] suchthat, for each i ∈ {1, . . . , k}, the entry of D2 corresponding to bi is θi. Indeed,[Ir|D2] can be obtained from [Ir|D1] by a sequence of row and column scalings.

Proposition 7.0.16. The Fano plane F7 is representable over a field F if andonly if the characteristic of F is 2.

Proposition 7.0.17. The non-Fano plane F−7 is representable over a field F ifand only if the characteristic of F is not 2.

Every matroid that is representable over some field is representable over somefinite field. A matroid M is F-representable if and only if all its minors are F-representable.

Definition 7.0.18. The minor-minimal matroids that are not F-representable arecalled the excluded/forbidden minors for F-representability. They are the non-F-representable matroids for which every proper minor is F-representable.

Lemma 7.0.19. If a matroid M is an excluded minor for representability over afield F, then so is its dual M∗.

Definition 7.0.20. If M is representable over F2, we say M is binary. If M isrepresentable over F3, we say M is ternary. Matroids representable over allfields are called regular/unimodular.

42


Proposition 7.0.21. Let k be an integer exceeding one. ThenU2,k is F-representableif and only if |F| ≥ k − 1.

Corollary 7.0.22. The matroidsU2,q+2 andUq,q+2 are excluded minors forGF (q)-representability.

Theorem 7.0.23. Let q be a power of a prime p. When 2 ≤ r ≤ p, the matroidUr,n is GF (q)-representable if and only if n ≤ q + 1.

Theorem 7.0.24. Let M be a simple rank r matroid representable over the fieldF. Then M is a submatroid of the projective geometry PG(r − 1,F).

Theorem 7.0.25. A matroid is binary if and only if it has no U2,4-minor.

Theorem 7.0.26. A matroid is ternary if and only if it has no minor isomorphicto any of the matroids U2,5, U3,5, F7, and F ∗7 .

Theorem 7.0.27. A matroid is quaternary if and only if it has no minor isomor-phic to any of the matroids U2,6, U4,6, P6, F

−7 , (F

−7 )∗, P8 and P=

8 .

Theorem 7.0.28. Let F be a field. If it is infinite, then the set of excluded minorsfor F-representability is infinite.

Theorem 7.0.29. Let q be a power of a prime p. When 2 ≤ r ≤ p, the matroidUr,n is GF (q)-representable if and only if 2 ≤ r ≤ p.

Theorem 7.0.30. A matroid is regular if and only if it has no minor isomorphicto any of the matroids U2,4, F7, and F ∗7 .

Definition 7.0.31. Let K be an extension field of F. An element u ∈ K is said tobe algebraic over F if u is a root of some non-zero polynomial in F[x]. If u is notalgebraic over F, it is transcendental over F. If every element of K is algebraicover F, then K is an algebraic extension of F, otherwise K is a transcendentalextension of F.

Definition 7.0.32. Let {t1, t2, . . . , tn} be a subset of K. An element s ∈ Kis algebraically dependent on {t1, t2, . . . , tn} over F if s is algebraic overF(t1, t2, . . . , tn). The latter occurs if and only if s is a root of an equation ofthe form

a0(t1, . . . , tn)xm + a1(t1, . . . , tn)xm−1 + · · ·+ am(t1, . . . , tn) = 0

where each ai(t1, . . . , tn) is a polynomial in t1, . . . , tn with coefficients in F,and at least one of these polynomials is non-zero. A finite subset T of K isalgebraically dependent over F if, for some t ∈ T , the element t is algebraicallydependent on T − t. If T is not algebraically dependent over F, it is calledalgebraically independent over F.

Theorem 7.0.33. Suppose that K is an extension field of a field F and E is afinite subset of K. The the collection I of subsets of E that are algebraicallyindependent over F is the set of independent sets of a matroid on E.

43


Corollary 7.0.34. Let T and U be finite subsets of an extension field K of a fieldF and suppose that U is algebraically independent over F.

i) If every element of T is algebraic over F, then U is algebraically indepen-dent over F(T ).

ii) T ∪U is algebraically independent over F if and only if T is algebraicallyindependent over F(U).

Definition 7.0.35. Let M be an arbitrary matroid, and (E, I) the matroid in theprevious extension field. Suppose that there is a map φ from E(M) into E suchthat, for all subsets T ⊂ E(M), the set T is independent in M if and only if|φ(T )| = |T | and φ(T ) is independent in (E, I). Then the matroid M is saidto be algebraic over F, and the map φ is called an algebraic representationof M over F. An algebraic matroid is one that is algebraic over some field.To emphasize the distinction between this representability and the previous, werefer to an F-representable matroid as being linearly representable over F.

Proposition 7.0.36. Let {t1, . . . , tn} be a non-empty subset of an extension fieldK of a field F. Then {t1, . . . , tn} is algebraically independent over F if and onlyif there is no non-zero polynomial f(x1, . . . , xn) with coefficients in F such thatf(t1, . . . , tn) = 0.

Proposition 7.0.37. If a matroid M is linearly representable over a field F, thenM is algebraic over F.

Proposition 7.0.38. If a matroid M is algebraic over a field F of character-istic zero, then M is linearly representable over F(T ) for some finite set T oftranscendentals over F.

Proposition 7.0.39. If a matroid M is algebraic over an extension field F(t) ofa field F, then M is algebraic over F.

Proposition 7.0.40. If a matroid M is algebraic over a field F and T ⊂ E(M),then M/(E − T ) is algebraic over F.

Proposition 7.0.41. If a matroid M is algebraic over a field F, then every minorof M is algebraic over F.

Definition 7.0.42. The operation of adding a point p freely to a specified flat Fin a matroid M is called a free extension, and is usually expressed as M +F p.This operation reverses deletion: (M +F p)− p = M .

Theorem 7.0.43. The non-Pappus matroid is not representable over any field.

Lemma 7.0.44. Suppose M is representable over the field F and N is obtainedfrom M by some sequence of deletions and contractions. Then N is also repre-sentable over F.

Proposition 7.0.45. Let P = {0, 2, 3, 5, . . .} be the collection of all possiblefield characteristics. Suppose M1 and M2 are matroids on disjoint ground sets,

44


with M1 only representable over field characteristics A1 ⊂ P and M2 onlyrepresentable over field characteristics A2 ⊂ P , with A1 ∩ A2 = ∅. Then thedirect sum M1 ⊕M2 is not representable over any field.

Definition 7.0.46. Let M be a matroid and let P = {0, 2, 3, 5, . . .} be the col-lection of all possible field characteristics. Then the characteristic set χ(M) isthe subset of P consisting of all characteristics over which M is representable.

Theorem 7.0.47. Let M be a simple rank r matroid representable over the fieldF. Then M is a submatroid of the projective geometry PG(r − 1,F).

Theorem 7.0.48. Let p be prime. Then the projective plane PG(2, p) is onlyrepresentable as a matroid over fields of characteristic p, i.e., χ(PG(2, p)) ={p}.

Corollary 7.0.49. Let p be prime and r ≥ 2. Then the projective geometryPG(r, p) is only representable as a matroid over fields of characteristic p, i.e.,χ(PG(r, p)) = {p}.

Theorem 7.0.50. If p > 3 is prime, then the affine plane AG(2, p) is only repre-sentable over fields of characteristic p, i.e., χ(AG(2, p)) = {p}.

Theorem 7.0.51. The following statements are equivalent for a matroid M :i) M is regular.ii) M is representable over every field.iii) M is binary and, for some field F of characteristic other than two, M is

F-representable.

Proposition 7.0.52. Let M be a binary matroid and F be a field. Then all F-representations ofM are projectively equivalent, soM is uniquely F-representable.

Theorem 7.0.53. A matroid is regular if and only if it has no minor isomorphicto any of the matroids U2,4, F7, and F ∗7 .

Definition 7.0.54. Let X and Y be flats in a matroid M . Then (X, Y ) is amodular pair of flats if

r(X) + r(Y ) = r(X ∪ Y ) + r(X ∩ Y ).

If Z is a flat of M such that (Z, Y ) is a modular pair for all flats Y , then Zis called a modular flat of M . Evidently E(M), cl(∅), and all rank-1 flats ofM are modular flats; so are all separators of M . Every projective geometryPG(n, q) has the property that all its flats are modular. A flat X is modular ina matroid M if and only if, in si(M), the flat corresponding to X is modular. Amatroid in which evey flat is modular, is called a modular matroid.

Proposition 7.0.55. A matroid M is modular if and only if, for every connectedcomponent N of M , the simple matroid associated with N is either a free ma-troid or a finite projective geometry.

45


Proposition 7.0.56. The following statements are equivalent for a flat X in amatroid M :

i) X is modular.ii) r(X) + r(Y ) = r(X ∪ Y ) for all flats Y that meet X in cl(∅).iii) r(X) + r(Y ) = r(M) for all compliments Y of X .

Recall that a line in a matroid is a rank-2 flat.

Corollary 7.0.57. In a loopless matroid, a hyperplane is modular if and onlyif it intersects every line; and a line is modular if and only if it intersects everyhyperplane.

Proposition 7.0.58. Let X be a modular flat in a matroid M and T be a subsetof E(M) contatining X . Then X is a modular flat of M |T

Corollary 7.0.59. If M is a simple matroid representable over GF (q), and X isa subset of E such that M |X ∼= PG(k − 1, q) for some k, then X is a modularflat of M .

Proposition 7.0.60. LetX be a modular flat in a matroidM and Y be a modularflat in M |X . Then Y is a modular flat in M .

Corollary 7.0.61. If X and Y are modular flats in a matroid M , then X ∩ Y isa modular flat.

Theorem 7.0.62. (Modular short-circuit axiom) The following statements areequivalent for a non-empty set X of elements in a simple matroid M :

i) X is a modular flat of M .ii) For every circuit C such that C −X is non-empty, there is an element x of

X such that (C −X) ∪ x is dependent.iii) For every circuit C and every element e of C −X , there is an element f

of X and a circuit C ′ such that e ∈ C ′ ⊂ f ∪ (C −X).

Corollary 7.0.63. In a simple matroid M , let X be a flat and X1, X2, . . . , Xk

be the components of M |X . Then X is modular iff M has distinct componentsM1, . . . ,Mk such that Xi is a modular flat of Mi for all i. In particular, modularflats of simple connected matroids are connected.

46

Chapter 8

Constructions

Definition 8.0.1. If a matroid M is obtained from a matroid N by deleting anon-empty subset T of E(N), then N is called an extension of M . A term thatis sometimes used instead of ’single-element extension’ is ’addition’. If N∗ is anextension of M∗ then N is called a coextension of M . In this case M = N/Tfor some T of E(N). If N is a series extension of M , then N is actually acoextension of M rather than an extension. We shall still say ’series extension’instead of the more accurate ’series coextension’.

Lemma 8.0.2. Let N be an extension of a matroid M by an element e and letM be the set of flats F of M such that F ∪ e is a flat of N having the same rankas F . ThenM has the following properties:

i) If F ∈M and F ′ is a flat of M containing F , then F ′ ∈M.ii) If F1, F2 ∈M and (F1, F2) is a modular pair, then F1 ∩ F2 ∈M.

Definition 8.0.3. An arbitrary setM of flats of a matroidM is called a modularcut if it satisfies (i) and (ii) of the lemma.

By the lemma, every single-element extension of a matroid gives rise to a mod-ular cut. The next result extablishes that every modular cut gives rise to a uniqueextension.

Theorem 8.0.4. LetM be a modular cut of a matroid M on a set E. Then thereis a unique extension N of M on E ∪ e such thatM consists of those flats F ofM for which F ∪ e is a flat of N having the same rank as F . Moreover, for allsubsets X of E,

rN(X) = rM(X)

47

CHAPTER 8. CONSTRUCTIONS

and

rN(X ∪ e) =

{rM(X), if clM(X) ∈M,

rM(X) + 1, if clM(X) 6∈ M.

Since there is a one-to-one correspondence between single element extensionsand modular cuts, we often refer to an extension as being determined by the cor-responding modular cut. If M = N\e andM is the modular cut correspondingto the extension N , we shall often write N as M +M e.

Corollary 8.0.5. The flats of M +M e fall into three disjoint classes:i) flats F of M that are not inM;ii) sets F ∪ e where F is a flat of M that is inM; andiii) sets F ∪ e where F is a flat of M that is not inM, and F is not contained

in a member F ′ ofM of rank r(F ) + 1.

Definition 8.0.6. If F is a flat of M , then the set MF of flats containing F isa modular cut. We callMF a principal modular cut and the extension deter-mined byMF a principal extension. We shall denote this principal extensionby M +F e and we say that the element e has been freely added to F or, ifF = E(M), that e has been freely added to M . In the last case, e is said to befree in M +E(M) e and this matroid is called the free extension of M . Whenadding the element freely to the flat F , we add it so that the only new circuitscreated are those which are forced by the fact that e has been placed on F . Everysuch circuit spans F .

Proposition 8.0.7. Let F be a flat of a matroid M and N = M +F e. Then

I(N) = I(M) ∪ {I ∪ e : I ∈ I(M) and clM(I) 6⊃ F}.

Definition 8.0.8. If adding e to each of the flats F1, . . . , Fm, the modular cutcorresponding to such an extension must contain {F1, . . . , Fm}. The smallestmodular cut containing this set which is the intersection of all modular cutscontaining it, is called the modular cut generated by {F1, . . . , Fm}.

Definition 8.0.9. A modular cut of a matroid M will be called proper if it is notequal to the set of all flats of M . LetM be such a modular cut, and nonempty,the elementary quotient of M with respect toM is the matroid (M +M e)/e.In general, a matroid Q is a quotient of M if there is a matroid N and a subsetX of E(N) such that M = N\X and Q = N/X . Thus if Q is a quotient of M ,then E(Q) = E(M)

Proposition 8.0.10. Let M1 and M2 be matroids. Then M1 is a quotient of M2

if and only if M∗2 is a quotient of M∗

1 .

Corollary 8.0.11. If Q is a quotient of M and r(Q) = r(M), then Q = M .

48


Proposition 8.0.12. A matroid Q is a quotient of M with r(M) − r(Q) = kif and only if there is a sequence M0, . . . ,Mk of matroids with M0 = M andMk = Q such that, for all i ∈ {1, . . . , k}, the matroid Mi is an elementaryquotient of Mi−1.

Proposition 8.0.13. Let M1 and M2 be matroids having rank functions r1 andr2, closure operators cl1 and cl2, and a common ground set E. The followingare equivalent:

i) M2 is a quotient of M1.ii) Every flat of M2 is a flat of M1.iii) If X ⊂ Y ⊂ E, then r1(Y )− r1(X) ≥ r2(Y )− r2(X).iv) Every circuit of M1 is a union of circuits of M2.v) If X ⊂ E, then cl1(X) ⊂ cl2(X).

Definition 8.0.14. A matroid M1 is a lift of a matroid M2 if M2 is a quotient ofM1. If M2 is an elementary quotient of M1, then M1 is an elementary lift ofM2. Thus M1 is a lift of M2 if there is a matroid N and a subset Y of E(N) suchthat N\Y = M1 and N/Y = M2. Hence M1 is a lift of M2 if and only if M∗

1 isa quotient of M∗

2

Definition 8.0.15. If N is the coextension of M by e in which e is a non-loopelement that is in every dependent flat, then N is called the free coextension ofM .

Proposition 8.0.16. Let e be an element of a matroid N and suppose that e isnot a loop. Then e is in every dependent flat of N if and only if e is free in N∗.

Definition 8.0.17. For r(M) > 0, the truncation T (M) of M is the elementaryquotient corresponding to the free extension, that is T (M) = (M +E(M) e)/e.I.e., we freely add an element to M and then contract it out. When r(M) = 0 wetake T (M) to be M . For all positive integers i, the ith truncation T i(M) of Mis defined inductively by T i(M) = T (T i−1(M)) where T 0(M) = M .

Proposition 8.0.18. Let M be a matroid of non-zero rank and let i be a non-negative integer not exceeding r(M). Then

I(T i(M)) = {X ∈ I(M) : |X| ≤ r(M)− i}.

Definition 8.0.19. For a matroid M1 on a set E1 and a matroid M2 on E2, with|E1| = |E2|, a bijective map φ : E1 → E2 is a strong map if φ−1(F ) is a flatof M1 for every flat F of M2. We say that a bijection φ is a weak map from M1

to M2 if φ−1(I) is independent in M1 for every independent set I of M2. In thiscase, M2 is called a weak-map image of M1.

Proposition 8.0.20. Let M1 and M2 be matroids and let φ : E1 → E2 be abijection. The following are equivalent:

i) The function φ is a weak map from M1 to M2.ii) If D is dependent in M1, then φ(D) is dependent in M2.

49


iii) If C is a circuit of M1, then φ(C) contains a circuit of M2.iv) If X ⊂ E1, then r1(X) ≥ r2(φ(X)).

Corollary 8.0.21. Suppose that M1 and M2 are matroids on E1 and E2 and thatφ : E1 → E2 is a bijection. If φ is a strong map, then φ is a weak map.

Corollary 8.0.22. If φ is a weak map from M1 to M2, then φ is a weak map fromM∗

1 to M∗2 .

Corollary 8.0.23. Suppose that φ : E1 → E2 and ψ : E2 → E1 are bijections.If both φ and ψ are weak maps, then M1

∼= M2.

Definition 8.0.24. The collection E of matroids on a fixed set E can be partiallyordered by taking M1 ≥ M2 if the identity map on E is a weak map from M1 toM2. Under this partial order, the weak order on E , the set has both a one anda zero, namely the free matroid on E and the rank-0 matroid on E. We say thatM1 is freer than M2 if M1 ≥M2.

50

Chapter 9

The Splitter Theorem

Definition 9.0.1. Let N be a class of matroids that is closed under minors andunder isomorphism. A member N of N is a splitter for N if, whenever Mis a member of N having an N -minor, either M ∼= N , or M has a 1-, or 2-separation. Thus N is a splitter for N if and only if N has no 3-connectedmember having a proper N -minor.

Theorem 9.0.2 (The Splitter Theorem). Let N be a non-empty, connected, sim-ple, cosimple minor of a 3-connected matroid M . Suppose that if N is a wheel,then M has no larger wheel as a minor, while if N is a whirl, M has no largerwhirl as a minor. Then either M = N , or M has a connected, simple, cosimpleminor M1 such that some single-element deletion or some single-element con-traction of M1 is isomorphic to N . Moreover, if N is 3-connected, so too isM1.

The first restatement of the theorem is the following:

Theorem 9.0.3. Let N be a class of matroids that is closed under minors andunder isomorphism. Let N be a 3-connected member of N having at least fourelements such that if N is a wheel, it is the largest wheel in N , while if N is awhirl, it is the largest whirl inN . Suppose there is no 3-connected member ofNthat has N as a minor and has one more element than N . Then N is a splitterfor N , that is, no 3-connected member of N has N as a proper minor.

The second reformulation in the case that N is 3-connected:

Theorem 9.0.4. Let M and N be 3-connected matroids such that N is a minorofM with at least four elements and ifN is a wheel, thenM has no larger wheelas a minor, while if N is a whirl, then M has no larger whirl as a minor. Thenthere is a sequence M0,M1, . . . ,Mn of 3-connected matroids with M0

∼= Nand Mn = M such that Mi is a single-element deletion or a single-elementcontraction of Mi+1 for all i ∈ {0, 1, . . . , n− 1}.

Recall that the smallest whirl W2 is isomorphic to U2,4 while the smallest3-connected wheel, M(W3) is isomorphic to M(K4). A third reformulation isthe following:

51

CHAPTER 9. THE SPLITTER THEOREM

Theorem 9.0.5. Let N be a class of matroids that is closed under isomorphismand under minors and letN be a 3-connected memeber ofN having at least fourelements such that no 3-connected member of N with an N minor has exactly|E(N)| + 1 elements. Suppose that if N is a wheel, then N contains no largerwheels, while if N is a whirl, N contains no larger whirls. Then every matroidinN can be obtained by a sequence of direct sums and 2-sums from copies of Nand members of N having no N -minor.

Corollary 9.0.6. The following are equivalent for a non-empty 3-connected ma-troid M .

i) For every element e of M , neither M\e nor M/e is 3-connected.ii) M has rank at least three and is isomorphic to a wheel or a whirl.

52

Chapter 10

Axiom Tables

Matroid terms interpreted for graphs

Matroid Term Symbol Graph interpretation

Independent Sets I Edges of forestsBases B Edges of spanning forests.Circuits C Edges of cycles.Rank r r(A) is the number of edges

of a spanning forest in A.Flats F Edges F for which there is a partition

Π of the vertices so that e ∈ Fwhenever e joins two vertices ofthe same block of Π.

Cocircuits C∗ Minimal edge cut-setsHyperplanes H Cocircuit complements or maximal flats.Closure A A contains edge whose endpoints

are connected by a path in A.

53

CHAPTER 10. AXIOM TABLES

For matrices

Matroid Term Symbol Matrix description

Independent sets I Linearly independent sets of columns.Bases B Maximal independent sets.Circuits C Minimal linearly dependent sets.Rank r Rank of the corresponding submatrix.Flats F Sets equal to their linear span.Hyperplanes H Corank 1 flats.Closure A Linear span.Cocircuits C∗ Complements of hyperplanes.Spanning sets S Subsets of columns whose linear span

contains all the columns of the matrix.

For transversal matroids

Matroid Term Symbol Description

Independent sets I MatchingsBases B Maximal matchingsCircuits C Minimal sets without a matchingRank r r(A) is size of a matching contained in A

For affine geometry

Matroid Term Interpretation in AG(n, q)

Points Ordered n-tuples in Fnq .

Independent sets {P1, . . . , Pn} is affinely independent if there are no non-trivialscalars ai such that

∑ki=1 ai = 0 and

∑ki=1 aiPi = 0.

Closure Affine closure: for S = {P1, . . . , Pn}, Q ∈ S if there arenon-trivial scalars ai such that

∑ni=1 ai = 1 and Q =

∑ni=1 aiPi.

Flats Affine flats: translates of subspaces of Fnq . Equiv. all x ∈ Fn

q

satisfying Ax = Au for some (n− r)× n A and u ∈ Fnq .

Hyperplanes Maximal flats: {(x1, . . . , xn) :∑n

i=1 aixi = b for ai, b ∈ Fq}.Rank r(AG(n, q)) = n+ 1, and r(S) = dim(S) + 1.

54


For PG(n, q)

Matroid Concept Interpretation in PG(n, q)

Points Lines through (0, . . . , 0) in AG(n+ 1, q). Equivalentlyequivalence classes [v] for 0 6= v ∈ Fn+1

q , wherev ∼ v′ if v = k · v′ for nonzero scalar k.

Independent sets Lin. ind: {P1, . . . , Pn} is indep if there are no non-trivial scalars aisuch that

∑ki=1 aiPi = 0.

Closure Lin. closure: Q ∈ S if there are non-trivial scalars aisuch that Q =

∑ni=1 aiPi.

Flats The points of F correspond to a subspace of Fn+1q . Equiv, a k-flat

consists of all vectors v satisfying Av = 0 forsome (n− k)× n matrix A.

Hyperplanes Maximal flats: {(x0, . . . , xn) :∑n

i=0 aixi = 0 for ai ∈ Fq}.Rank r(S) is the matrix rank of the matrix whose column vectors

are the points of S. r(PG(n, q)) = n+ 1.

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Matroid Theory - WordPress.com...2017/09/01 · Matroid Theory Travis Dirle May 30, 2017 2 Contents 1 Basics and Deﬁnitions 1 2 Duality 9 3 Minors 11 4 Connectivity 19 5 Graphic