graph2 - basics.sjtu.edu.cnbasics.sjtu.edu.cn/~chen/teaching/GR08/graph2.pdf · GRAPH THEORY (II) Page 2 1.4 Connectivity A non-emptygraph Gis called connectedif any two of its vertices

Page 1

GRAPH THEORY

Yijia ChenShanghai Jiaotong University

2008/2009

Shanghai

GRAPH THEORY (II) Page 2

1.4 Connectivity

Shanghai


1.4 Connectivity

A non-empty graph G is called connected if any two of its vertices are linked by a path in G.

Shanghai


1.4 Connectivity


If U ⊆ V (G) and G[U ] is connected, we also call U itself connected (in G).

Shanghai


1.4 Connectivity



Instead of ‘not connected’ we usually say ‘disconnected’.

Shanghai


1.4 Connectivity



Instead of ‘not connected’ we usually say ‘disconnected’.

Proposition. The vertices of a connected graph G can always be enumerated, say as

v1, . . . , vn, so that Gi := G[v1, . . . , vi] is connected for every i.

Shanghai


Let G = (V, E) be a graph. A maximal connected subgraph of G is called a component of

G.

Shanghai


Let G = (V, E) be a graph. A maximal connected subgraph of G is called a component of

G.

note: A component, being connected, is always non-empty: the empty graph, therefore, has

no components.

Shanghai


Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if

V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.

If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge

from X , we say that X separates A and B in G.

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.



note: This implies A ∩ B ⊆ X

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.




The unordered pair {A, B} is a separation of G if A ∪ B = V (G) and G has no edge

between A \ B and B \ A.

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.






Clearly, the latter is equivalent to saying that A ∩ B separates A from B.

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.







If both A \ B and B \ A are non-empty, the separation is proper.

Shanghai



V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.







If both A \ B and B \ A are non-empty, the separation is proper.

The number |A ∩ B| is the order of the separation {A, B}.

Shanghai


If U is any set of vertices (usually of G), we write G − U for G[V \ U ].

Shanghai



In other words, G − U is obtained from G by deleting all the vertices in U ∩ V and their

incident edges.

Shanghai



In other words, G − U is obtained from G by deleting all the vertices in U ∩ V and their

incident edges.

We say that X separates G if G − X is disconnected, that is, if X separates in G some two

vertices that are not in X .

Shanghai


G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set

X ⊆ V with |X | < k.

Shanghai




In other words, no two vertices of G are separated by fewer than k other vertices.

Shanghai





Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the

non-trivial connected graphs.

Shanghai







The greatest integer k such that G is k-connected is the connectivity κ(G) of G.

Shanghai








For nonempty G, κ(G) = 0 if and only G is disconnected or a K1.

Shanghai








For nonempty G, κ(G) = 0 if and only G is disconnected or a K1.

κ(Kn) = n − 1 for all n ≥ 1.

Shanghai


For a subset F of [V ]2 we write G − F := (V, E \ F ) and G + F := (V, E ∪ F ).

Shanghai



If |G| > 1 and G − F is connected for every set F ⊆ E of fewer than � edges, then G is

called �-edge-connected.

Shanghai





The greatest integer � such that G is �-edge-connected is the edge-connectivity λ(G) of G.

If particular, we have λ(G) = 0 if G is disconnected.

Shanghai





The greatest integer � such that G is �-edge-connected is the edge-connectivity λ(G) of G.

If particular, we have λ(G) = 0 if G is disconnected.

Proposition. If G is non-trivial then κ(G) ≤ λ(G) ≤ δ(G).

Shanghai


High connectivity requires a large minimum degree. Conversely, large minimum degree

does not ensure high connectivity, not even high edge-connectivity.

Shanghai


High connectivity requires a large minimum degree. Conversely, large minimum degree

does not ensure high connectivity, not even high edge-connectivity.

Theorem.[Mader 1972] Let 0 �= k ∈ N. Every graph G with d(G) ≥ 4k has a(k + 1)-connected subgraph H such that ε(H) > ε(G) − k.

Shanghai


Proof.

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).

Consider the subgraphs G′ ⊆ G such that

|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).


|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)

Let H be one of the smallest order.

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).


|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)

Let H be one of the smallest order. No G′ as in (1) can have order exactly 2k; otherwise

‖G′‖ > γk ≥ 2k2 >

(|G′|2

).

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).


|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)


‖G′‖ > γk ≥ 2k2 >

(|G′|2

).

The minimality of H then implies δ(H) > γ; otherwise we could delete a vertex of degree

at most γ and obtain a graph G” of smaller order satisfying (1).

Shanghai


Proof.

Let γ := ε(G)(≥ 2k).


|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)


‖G′‖ > γk ≥ 2k2 >

(|G′|2

).

The minimality of H then implies δ(H) > γ; otherwise we could delete a vertex of degree

at most γ and obtain a graph G” of smaller order satisfying (1).

Hence |H| ≥ γ, thus ‖H‖ > γ(|H| − k) implies

ε(H) > γ − k.

Shanghai


Proof. [Cont’d]

Shanghai


Proof. [Cont’d]

It remains to show that H is (k + 1)-connected.

Shanghai


Proof. [Cont’d]


If not, then H has a proper separation {U1, U2} of order at most k.

Shanghai


Proof. [Cont’d]


If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for

i = 1, 2.

Shanghai


Proof. [Cont’d]



i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in

H1, we have |H1| ≥ γ ≥ 2k.

Shanghai


Proof. [Cont’d]




H1, we have |H1| ≥ γ ≥ 2k. Similarly, |H2| ≥ 2k.

Shanghai


Proof. [Cont’d]





By the minimality of H with respect to (1), we have

‖Hi‖ ≤ γ(|Hi| − k)

for i = 1, 2.

Shanghai


Proof. [Cont’d]






‖Hi‖ ≤ γ(|Hi| − k)

for i = 1, 2. But then

‖H‖ ≤ ‖H1‖ + ‖H2‖≤ γ(|H1| + |H2| − 2k)

≤ γ(|H| − k) (as |H1 ∩ H2| ≤ k),

Shanghai


Proof. [Cont’d]






‖Hi‖ ≤ γ(|Hi| − k)

for i = 1, 2. But then

‖H‖ ≤ ‖H1‖ + ‖H2‖≤ γ(|H1| + |H2| − 2k)

≤ γ(|H| − k) (as |H1 ∩ H2| ≤ k),

which contradicts (1) for H . �

Shanghai


1.5 Trees and forests

Shanghai



An acyclic graph, one not containing any cycles, is called a forest.

Shanghai




A connected forest is called a tree.

Shanghai





The vertices of degree 1 in a tree are its leaves,

Shanghai





The vertices of degree 1 in a tree are its leaves, except that the root of a tree is never called a

leaf, even if it has degree 1.

Shanghai


Theorem. The following assertions are equivalent for a graph T :

Shanghai



(i) T is a tree;

Shanghai



(i) T is a tree;

(ii) Any two vertices of T are linked by a unique path in T ;

Shanghai



(i) T is a tree;


(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge

e ∈ T ;

Shanghai



(i) T is a tree;



e ∈ T ;

(iv) T is maximally acyclic, i.e. T contains no cycle but T + xy does, for any two

non-adjacent vertices x, y ∈ T .

Shanghai



(i) T is a tree;



e ∈ T ;



We write xTy for the unique path in a tree T between two vertices x and y.

Shanghai



(i) T is a tree;



e ∈ T ;



We write xTy for the unique path in a tree T between two vertices x and y.

Every connected graph contains a spanning tree: by the equivalence of (i) and (iii), any

minimal connected spanning subgraph will be a tree.

Shanghai


Corollary. The vertices of a tree can always be enumerated, say as v1, . . . , vn, so that every

vi with i ≥ 2 has a unique neighbour in {v1, . . . , vi−1}.

Shanghai




Corollary. A connected graph with n vertices is a tree if and only if it has n − 1 edges.

Shanghai




Corollary. A connected graph with n vertices is a tree if and only if it has n − 1 edges.

Corollary. If T is a tree and G is any graph with δ(G) ≥ |T | − 1, then T ⊆ G, i.e., G has a

subgraph isomorphic to T .

Shanghai


Sometimes it is convenient to consider one vertex of a tree as special;

Shanghai


Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then

called the root of this tree.

Shanghai




A tree T with a fixed root r is a rooted tree.

Shanghai





Writing x ≤ y for x ∈ rTy then defines a partial ordering on V (T ), the tree-order

associated with T and r.

Shanghai







We shall think of this ordering as expressing height: if x < y we say that x lies below y in

T .

Shanghai







We shall think of this ordering as expressing height: if x < y we say that x lies below y in

T .

We call

y� :={x | x ≤ y

}and �x :=

{y | y ≥ x

}

the down-closure of y and the up-closure of x.

Shanghai


Shanghai


- The root r is the least element in this partial order.

Shanghai



- The leaves of T are its maximal elements.

Shanghai




- The ends of any edge of T are comparable.

Shanghai





- The down-closure of every vertex is a chain, a set of pairwise comparable elements.

Shanghai





- The down-closure of every vertex is a chain, a set of pairwise comparable elements.

- The vertices at distance k from r have height k and form the kth level of T .

Shanghai


Given a graph H , we call a path P an H-path if P is non-trivial and meets H exactly in its

ends.

Shanghai



ends.

In particular, the edge of any H-path of length 1 is never an edge of H .

Shanghai



ends.


A rooted tree T contained in a graph G is called normal in G if the ends of every T -path in

G are comparable in the tree-order of T .

Shanghai



ends.


A rooted tree T contained in a graph G is called normal in G if the ends of every T -path in

G are comparable in the tree-order of T .

If T spans G, this amounts to requiring that two vertices of T must be comparable whenever

they are adjacent in G.

Shanghai


Lemma. Let T be a normal tree in G.

Shanghai



(i) Any two vertices x, y ∈ T are separated in G by the set x� ∩ y�.

Shanghai



(i) Any two vertices x, y ∈ T are separated in G by the set x� ∩ y�.

(ii) If S ⊆ V (T ) = V (G) and S is down-closed, then the components of G − S are spanned

by the sets �x with x minimal in T − S.

Shanghai


Normal spanning trees are also called depth-first search trees.

Shanghai


Normal spanning trees are also called depth-first search trees.

Proposition. Every connected graph contains a normal spanning tree, with any specified

vertex as its root.

Shanghai


1.6 Bipartite graphs

Shanghai



Let r ≥ 2 be an integer. A graph G = (V, E) is called r-partite if V admits a partition into r

classes such that every edge has its ends in different classes:

Shanghai




classes such that every edge has its ends in different classes: vertices in the same partition

class must not be adjacent.

Shanghai






Instead of ‘2-partite’ one usually says bipartite.

Shanghai






Instead of ‘2-partite’ one usually says bipartite.

Theorem. A graph is bipartite if and only if it contains no odd cycle.

Shanghai


1.7 Contraction and minors

Shanghai



Let e = xy be an edge of a graph G = (V, E). By G/e we denote the graph obtained from

G by contracting the edge e into a new vertex ve, which becomes adjacent to all the former

neighbours of x and of y.

Shanghai



Let e = xy be an edge of a graph G = (V, E). By G/e we denote the graph obtained from

G by contracting the edge e into a new vertex ve, which becomes adjacent to all the former

neighbours of x and of y.

Formally, G/e is a graph (V ′, E′) with vertex set V ′ := (V \ {x, y})∪̇{ve} (where ve is the

‘new’ vertex, i.e., ve /∈ V ∪ E) and edge set

E′ :={

vw ∈ E∣∣ {v, w} ∩ {x, y} = ∅

}

∪{

vew∣∣ xw ∈ E \ {e} or yw ∈ E \ {e}

}.

Shanghai


More generally, if X is another graph and {Vx|x ∈ V (X)} is a partition of V into connected

subsets such that, for any two vertices x, y ∈ X , there is a Vx-Vy edge in G if and only if

xy ∈ E(X), we call G an MX and write G = MX .

Shanghai





The sets Vx are the branch sets of this MX .

Shanghai





The sets Vx are the branch sets of this MX .

Intuitively, we obtain X from G by contracting every branch set to a single vertex and

deleting any ‘parallel edges’ or ‘loops’ that may arise.

Shanghai


Proposition. G is an MX if and only if X can be obtained from G by a series of edge

contractions, i.e. if and only if there are graphs G0, . . . , Gn and edges ei ∈ Gi such that

G0 = G, Gn = X , and Gi+1 = Gi/ei for all i < n.

Shanghai


If G = MX is a subgraph of another graph Y , we call X a minor of Y and write X � Y .

Shanghai



- Every subgraph of a graph is also its minor; in particular, every graph is its own minor.

Shanghai




- Any minor of a graph can be obtained from it by first deleting some vertices and edges, and

then contracting some further edges.

Shanghai




- Any minor of a graph can be obtained from it by first deleting some vertices and edges, and

then contracting some further edges.

- Conversely, any graph obtained from another by repeated deletions and contractions (in

any order) is its minor.

Shanghai


If we replace the edges of X with independent paths between their ends (so that none of

these paths has an inner vertex on another path or in X), we call the graph G obtained a

subdivision of X and write G = TX .

Shanghai





If G = TX is a subgraph of another graph Y , then X is a topological minor of Y .

Shanghai






If G = TX , we view V (X) as a subset of V (G) and call these vertices the branch vertices

of G; the other vertices of G are its subdividing vertices.

Shanghai






If G = TX , we view V (X) as a subset of V (G) and call these vertices the branch vertices

of G; the other vertices of G are its subdividing vertices.

All subdividing vertices have degree 2, while the branch vertices retain their degree from X .

Shanghai


Proposition. (i) Every TX is also an MX ; thus, every topological minor of a graph is also

its (ordinary) minor.

Shanghai




(ii) If Δ(X) ≤ 3, then every MX contains a TX ; thus, every minor with maximum degree

at most 3 of a graph is also its topological minor.

Shanghai




(ii) If Δ(X) ≤ 3, then every MX contains a TX ; thus, every minor with maximum degree

at most 3 of a graph is also its topological minor.

Proposition. The minor relation � and the topological-minor relation are partial orderingson the class of finite graphs, i.e. they are reflexive, antisymmetric and transitive.

Shanghai


1.8 Euler tours

Shanghai


1.8 Euler tours

A walk (of length k) in a graph G is a non-empty alternating sequence

v0e0v1e1 . . . ek−1vk

of vertices and edges in G such that ei = {vi, vi+1} for all i < k.

Shanghai


1.8 Euler tours


v0e0v1e1 . . . ek−1vk


If v0 = vk, the walk is closed.

Shanghai


1.8 Euler tours


v0e0v1e1 . . . ek−1vk



If the vertices in a walk are all distinct, it defines an path in G.

Shanghai


1.8 Euler tours


v0e0v1e1 . . . ek−1vk



If the vertices in a walk are all distinct, it defines an path in G.

In general, every walk between two vertices contains a path between these vertices.

Shanghai


A closed walk in a graph is an Euler tour if it traverses every edge of the graph exactly once.

Shanghai



A graph is Eulerian if it admits an Euler tour.

Shanghai



A graph is Eulerian if it admits an Euler tour.

Theorem.[Euler 1736] A connected graph is Eulerian if and only if every vertex has even

degree.

Shanghai

Documents

graph2 - basics.sjtu.edu.cnbasics.sjtu.edu.cn/~chen/teaching/GR08/graph2.pdf · GRAPH THEORY (II) Page 2 1.4 Connectivity A non-emptygraph Gis called connectedif any two of its vertices