On the arrangement of cliques in Chordal graphs with respect to the cuts

Abstract

A cut (A, B) (where B = V - A) in a graph G = (V, B) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y ε B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with κ(G) + 1 vertices (where κ(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 ≤ i ≤ κ(G) + 1, there exists a clique Ki such that \K 2\ = κ(G) + 1, \A ∪ K,\ = i and \B ∪ K2\ = κ(G) + l-i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be ω(k 2), where κ(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least κ(G)(κ(G)+1)/2 where κ(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to κ(G). This result is tight.