Menger’s Theorem Part II

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Menger’s Theorem Part II. Graphs & Algorithms Lecture 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 ,

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## Menger’s Theorem Part II

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### Menger’s Theorem Part II

Graphs & Algorithms

Lecture 4

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|.

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| < 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 iffG is disconnected or K1
• (Kn) = n – 1 for all n¸ 1
• (Cn) = 2 for all n¸ 3
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)

Menger’s 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 Menger’s Theorem to the line graph L(G):

• the vertex set of L(G) is the edge set of G
• e, f2E(G) are adjacent in L(G) iff eÅf;
Menger’s 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.