@MISC{Chandran_cutsand, author = {L. Sunil Chandran and N. S. Narayanaswamy}, title = {Cuts and Connectivity in Chordal Graphs}, year = {} }

Share

OpenURL

Abstract

A cut (A; B) in a graph G is called internal, i N(A) 6= B and N(B) 6= A. In this paper, we study the structure of internal cuts in chordal graphs. We show that if (A; B) is an internal cut in a chordal graph, then for each i, 0 i (G)+1, there exists a clique K i such that jK i j = (G)+1, jK i T Aj = i and jK i T Bj = k+1 i, where (G) is the vertex connectivity of G. In general, there can be an exponential number of internal cuts in a chordal graph, while the number of maximal cliques can be at most n (G) 1 [2]. Also there exists chordal graphs, all of whose maximal cliques are of size (G) + 1. Thus, above result throws some light as to the way the cliques are arranged in chordal graphs, with respect to their cuts. We also show that in a chordal graph G, every internal cut should contain at least (G)((G)+1) 2 edges. This lower bound is tight, in the sense that there exists chordal graphs with internal cuts having exactly (G)((G)+1) 2 edges.