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Connectivity - Menger’s Theorem

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

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Connectivity - Menger’s Theorem

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

  2. Separators • Let G = (V, E) be a graph, A, B µV, and XµV • 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

  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 K1 • (Kn) = n – 1 for all n¸ 1 • (Cn) = 2 for all n¸ 3 • (Qd) = d for all d¸ 1 (Qd´d-dimensional hypercube)

  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 ifany two non-adjacent vertices of G are joined by k independent paths? • independent paths: pairwise internally vertex disjoint • Example of a 3-connected graph

  5. Menger’s Theorem Theorem (Menger, 1927)Let G = (V, E) be a graph and s and t distinct, non-adjacent vertices. Let • Xµ 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?)

  6. Menger’s Theorem II Theorem (multiple sources and sinks)Let G = (V, E) be a graph and S, TµV. Let • XµV be a set separating S from T of minimal size, • P be a set of disjointS – 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

  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 ): LemmaFor every e2E(G), we have (G – e) ¸(G) – 1.

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