1 / 6

Menger’s Theorem Part II

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 ,

vanida
Download Presentation

Menger’s Theorem Part II

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Menger’s Theorem Part II Graphs & Algorithms Lecture 4

  2. 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|.

  3. 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

  4. 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)

  5. 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;

  6. 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.

More Related