Download presentation

Presentation is loading. Please wait.

Published byReagan Foreman Modified over 6 years ago

1
Connectivity - Menger’s Theorem Graphs & Algorithms Lecture 3

2
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

3
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

4
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

5
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

6
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

7
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

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google