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Matroids, graphs in surfaces, and the Tutte polynomial 2016 International Workshop on Structure in Graphs and Matroids Iain Moatt and Ben Smith Royal Holloway, University of London Eindhoven, 29 th July 2016

Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

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Page 1: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

Matroids, graphs in surfaces, and theTutte polynomial

2016 International Workshop on Structure in Graphs andMatroids

Iain Moffatt and Ben Smith

Royal Holloway, University of London

Eindhoven, 29th July 2016

Page 2: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Overview

I Introduce matroidal analogues of various notions ofembedded graphs.

I Introduce by applications to the theory of the Tuttepolynomial:1. Extensions of the Tutte polynomial to graphs in

surfaces.2. Incomplete aspects of the theory.3. matroid model.4. Topological graphs ↔ matroid models

Page 3: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

2 Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A review of the Tutte polynomial

The Tutte polynomial, T(G;x,y)

Polynomial valued graph invariant, T : Graphs→ Z[x,y].

I Importance due to applications / combinatorial info.(colourings, flows, orientations, codes, Sandpile model,Potts & Ising models (statistical physics), QFT, Jones &homflypt polynomials (knot theory), ...)

Definition (deletion-contraction)

T(G;x,y) =

1 if G edgelessxT(G/e) if e a bridgeyT(G\e) if e a loopT(G\e) + T(G/e) otherwise

Page 4: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

2 Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A review of the Tutte polynomial

State sum formulation (T(G) is well-defined)

T(G) =∑A⊆E

(x− 1)r(G)−r(A)(y − 1)|A|−r(A)

where r(A) = #verts.−#cpts. of (V,A) = rank of A .

I T is defined for matroids (e.g., r= rank function).I T(C(G)) = T(G), where C(G) is cycle matroidI Matroids often ‘complete’ graph results (e.g.duality)

Page 5: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

3 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Graphs in surfaces

I Plane graph - drawn on a sphere, edges don’tmeet, faces are disks.

I Embedded graph = graph in surface - drawn onsurface, edges don’t meet.

I Cellularly embedded graph - drawn on surface,faces are disks.

Page 6: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

4 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A Topological Tutte polynomial

The Bollobás-Riordan-Krushkal polynomialK(G;x,y,a,b) :=

∑A⊆E(G)

xr(G)−r(A)y|A|−r(A)aγ(A)bγ∗(Ac)

γ(A) := Euler genus of nbhd. of subgraph of G on Aγ∗(Ac) := Euler genus of nbhd. of subgraph of G∗ on Ac

I T(G;x,y) = K(G;x− 1,y − 1,1,1)

I G plane graph =⇒ T(G;x,y) = K(G;x−1,y−1,a,b).

Page 7: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

4 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A Topological Tutte polynomial

I Deletion-contraction definition of the topologicalTutte polynomial:

Page 8: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

4 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A Topological Tutte polynomial

I Deletion-contraction definition of the topologicalTutte polynomial:

I No (full) recursive definition.

Page 9: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

4 Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

A Topological Tutte polynomial

I Deletion-contraction definition of the topologicalTutte polynomial:

I No (full) recursive definition.I =⇒ cell. embedded graphs are not the correctframework for the topological Tutte polynomial!

I What is the correct framework?

Page 10: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

5 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Look to matroids

I Why does deletion-contraction fail?

wants ribbon graph contraction

wants graph contraction

wants deletion as contraction in dual

¿ contract ?

I Exponents demand incompatible notions ofdeletion and contraction....

Page 11: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

5 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Look to matroids

I Why does deletion-contraction fail?

wants ribbon graph contraction

wants graph contraction

wants deletion as contraction in dual

¿ contract ?

Cycle matroid, C(G)

Bond matroid, B(G*)

Delta-matroid, D(G)

I Exponents demand incompatible notions ofdeletion and contraction....

I ...but these are provided by various matroids.

Page 12: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

6 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Delta-matroids

Symmetric Exchange Axiom (SEA): ∀X,Y ∈ F , if ∃u ∈ X4Y,then ∃v ∈ X4Y such that X4{u,v} ∈ F .

matroids (via bases)M = (E,B)

I B 6= ∅, subsets of EI B satisfies SEAI X,Y ∈ B =⇒ |X| = |Y|

Cycle matroid (trees)

M(G) = (E, {{2}, {3}})

delta-matroidsM = (E,F)

I F 6= ∅, subsets of EI F satisfies SEAI X,Y ∈ F =⇒ |X| = |Y|

∆-matroid (quasi-trees)

D(G) = (E, {{1,2,3}, {2}, {3}})

I Dmin = (E, {smallest sets}) a matroidI Dmax = (E, {biggest sets}) a matroidI D(G)min = C(G)I D(G)max = B(G∗) = (C(G∗))∗

Page 13: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

7 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

(matroid, delta-matroid, matroid)

I Associate triple to embedded graph:

I Generally, consider triples

(M,D,N) of (matroid, delta-matroid, matroid)

I Deletion & contraction:

(M,D,N)\e := (M\e,D\e,N\e), (M,D,N)/e := (M/e,D/e,N/e)

I Important observation: different actions of deletioncontraction,

(Dmin)/e 6= (D/e)min, (D\e)max 6= (Dmax\e).

(So we have more than the delta-matroid.)

Page 14: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

8 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

Strong maps and matroid perspectives

I There is structure we are not seeing.I Not all (graphic) triples can arise as minors of

(B(G∗),D(G),C(G)),I e.g., 12 triples (M,D,N) on 1 element, only 5 arise.I =⇒ missing conditions.

Matroid perspectivesA matroid perspective, is a pair of matroids (M,N) overE such that1. ⇐⇒ every circuit of M is union of circuits of N2. ⇐⇒ every flat of N is a flat of M,3. ⇐⇒ rM(B)− rM(A) ≥ rN(B)− rN(A) when A ⊆ B ⊆ E4. ⇐⇒ M = H\A and N = H/A, for some H on E t A.

I Examples of matroid perspectivesI (B(G∗),C(G))I (C(G),C(H)) where H from G by identifying verticesI (Dmax,Dmin) where D a delta-matroid

Page 15: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

9 A matroidal setting

Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

∆-perspectives

∆-perspectivesAn ∆-perspective is a triple (M,D,N) such that

1. M and N are matroids, and D is adelta-matroid over the same set,

2. (M,Dmax) is a matroid perspective3. (Dmin,N) is a matroid perspective

I Example: (B(G∗),D(G),C(G)) is a ∆-perspective.

TheoremIf (M,D,N) is an ∆-perspective, then so are (M,D,N)\eand (M,D,N)/e.

(M,D,N) from cell. embed. graph ; its minors are.

Page 16: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

10 Matroidpolynomials

Graphicalanalogues

Unifying TopologicalTutte polynomials

‘Tutte polynomial’ of perspectives

I There is a canonical way to construct ‘Tuttepolynomials’ of objects (via Hopf algebras).

Definition: Tutte polynomial of (M,D,N)

K(M,D,N) :=∑A⊆E

xr′(E)−r′(A)y|A|−r(A)aρ(A)−r′(A)br(A)−ρ(A),

where ρ = 12 (rmax + rmin).

I Theorems:I Contains Bollobás-Riordan-Krushkal polynomial

K(G;x,y,a,b) = bγ(G)K((M,D,N);x,y,a2,b−2)

I K(M,D,N) has a 6 term deletion-contraction relation.I duality formula, convolution formula, universality,...

I ∆-perspectives correct setting for topological Tuttepolynomials.

I Results that should hold for BRK-polynomial but donot, hold for the matroid version of the polynomial.

Page 17: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

11 Graphicalanalogues

Unifying TopologicalTutte polynomials

The graphical analogue

I Cellularly embedded graphs 6↔ ∆-perspectives.

I Pseudo-surface = surface withpinch points.

I Graph in pseudo surface - notnecessarily cell. embedded.

I Deletion and contraction defined in natural way:delete contract

∆-persps. ↔ graphs in pseudo-surfaces

I 7→ (B(G∗),D(G),C(G)) =: P(G)

I P(G)/e = P(G/e), P(G)\e = P(G\e), (P(G))∗ = P(G∗)

I Bollobás-Riordan-Krushkal polynomial is not apolynomial of cellularly embedded graphs.

I It is a polynomial of graphs in pseudo-surfaces.

Page 18: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

12 Graphicalanalogues

Unifying TopologicalTutte polynomials

The graphical analogue of subobjects

I Natural sub-objects of (M,D,N).I (M,D,N) ↔ graphsI (M,D,N) ↔ cell. embed. in surfacesI (M,D,N) ↔ cell. embed. in pseudo-surfacesI (M,D,N) ↔ non-cell. embed. in surfacesI (M,D,N) ↔ non-cell. embed. in pseudo-surfaces

I Concepts of minors, duals, etc. are compatible.

Page 19: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

13 Unifying TopologicalTutte polynomials

Three Topological Tutte polynomials

I Various candidates for the topological Tuttepolynomial in literature:

I M. Las Vergnas’ (1978), L(G;x,y, z)I B. Bollobás and O. Riordan’s (2001/2), R(G;x,y, z)I V. Kruskal’s (2011), K(G;x,y,a,b)

I Each corresponds to subobject

I =⇒ each polynomial is a topological Tuttepolynomial but for a different notion of embeddedgraph.

I Challenge: use this to find new combinatorialinterpretations!

Page 20: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

13 Unifying TopologicalTutte polynomials

Thank You!

Page 21: Matroids, graphs in surfaces, and the Tutte polynomial - 2016 … · 2016-07-28 · Definition(deletion-contraction) T(G;x;y) = 8 >> >< >> >: 1 ifGedgeless xT(G=e)

13

Tutte polynomial

Topologicalextensions

A matroidal setting

Matroidpolynomials

Graphicalanalogues

13 Unifying TopologicalTutte polynomials