Mengers Theorem Part II

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Mengers Theorem Part II

• Graphs Algorithms
• Lecture 4

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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.

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Edge-Connectivity Number ?(G)

• G is k-edge-connected if
• V(G) 2 and
• G X is connected for every set of edges X with
  X lt k.
• That is, no two vertices of G can be separated by
  less than k edges of G.
• G is 2-connected if and only if G is connected,
  contains at least 2 vertices and no bridge.
• Edge-connectivity number ?(G)the greatest
  integer k such that G is k-edge-connected
• ?(G) 0 iff G is disconnected or K1
• ?(Kn) n 1 for all n 1
• ?(Cn) 2 for all n 3

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Relation between ?(G) and ?(G)

• ?(G) and ?(G) can substantially deviate
• Example 2 cliques of size l sharing one
  vertex?(G) 1, ?(G) l 1
• PropositionEvery graph G on at least two
  vertices satisfies ?(G) ?(G) ?(G) .
• (?(G) minimum degree of G)

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Mengers Theorem IV

• Theorem (edge version)Let G (V, E) be a graph
  and s and t distinct vertices. Let
• X be a set of edges separating s from t of
  minimal size
• P be a set of pairwise edge disjoint s t paths
  of maximal size.
• Then we have X P.
• Proof
• Apply Mengers Theorem to the line graph L(G)
• the vertex set of L(G) is the edge set of G
• e, f 2 E(G) are adjacent in L(G) iff e Å f ?

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Mengers Theorem V

• Theorem (global edge version)A graph is
  k-edge-connected if and only if it contains k
  pairwise edge disjoint paths between any two
  distinct vertices.
• ProofFollows directly from the local version.