Connectivity - Menger

1
Connectivity - Mengers 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

3
Connectivity Number ?(G)

• G is k-connected if
• V(G) gt k and
• no set of vertices X with X lt 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
Mengers 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?)

6
Mengers 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 Mengers Theorem to s and t in this new
  graph

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Mengers 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 e 2 E(G), we have ?(G e)
  ?(G) 1.