Connectivity - Menger’s Theorem Graphs & Algorithms Lecture 3

Separators Let G = ( V, E ) be a graph, A, B µ V, and X µ V X separates A and B in G if every A - B path in G contains a vertex from X X is a separating set (vertex cut) of G if G – X is disconnected or contains just one vertex. Examples of separators – trees with at least 3 vertices: every vertex of degree ¸ 2 – bipartite graphs: any partition class – cliques of size l : every set of size l - 1 2

Connectivity Number  ( G ) G is k -connected if – | V ( G )| > k and – no set of vertices X with | X | < k separates G. G is 2-connected if and only if G is connected, contains at least 3 vertices and no articulation point. Connectivity number  ( G ): the greatest integer k such that G is k -connected –  ( G ) = 0 iff G is disconnected or K 1 –  ( K n ) = n – 1 for all n ¸ 1 –  ( C n ) = 2 for all n ¸ 3 –  ( Q d ) = d for all d ¸ 1 ( Q d ´ d -dimensional hypercube) 3

Structure of k -connected graphs Example: Blocks are 2-connected – maximal set of edges such that any two edges lie on a common simple cycle – every vertex is in a cycle – there are at least two independent (internally vertex disjoint) paths between any two non-adjacent vertices Is it true that a graph G is k -connected if and only if any two non-adjacent vertices of G are joined by k independent paths? – independent paths: pairwise internally vertex disjoint Example of a 3-connected graph 4

Menger’s Theorem Theorem (Menger, 1927) Let G = ( V, E ) be a graph and s and t distinct, non- adjacent vertices. Let X µ V \ {s, t} be a set separating s from t of minimum size, P be a set of independent s – t paths of maximum size. Then we have | X | = | P |. Clearly: | X | ¸ | P |. We need to show: | X | = | P |, i.e., there exist k :=| X | independent s – t paths. (Why is this not obvious?) 5

Menger’s Theorem II Theorem (multiple sources and sinks) Let G = ( V, E ) be a graph and S, T µ V. Let X µ V be a set separating S from T of minimal size, P be a set of disjoint S – T paths of maximal size. Then we have | X | = | P |. Proof insert two new vertices s and t into G connect s to all vertices of S and t to all vertices of T apply Menger’s Theorem to s and t in this new graph 6

Menger’s Theorem III Theorem (Whitney, 1932, global version) A graph is k -connected if and only if it contains k independent paths between any two distinct vertices. Proof ( : clear ) : Lemma For every e 2 E ( G ), we have  ( G – e ) ¸  ( G ) – 1. 7