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,E) is called internal, iff there exists a node x in A which is not adjacent to any node in B and there exists a node y ∈ B such that it is not adjacent to any node 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 nodes (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, for each i, 0 ≤ i ≤ κ(G) +1, there exists a clique K _i such that |K _i | = κ(G) + 1, |A ∩ K _i | = i and |B ∩ K _i | = κ(G) + 1 – 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 $\frac{\kappa(G)(\kappa (G)+1)}{2}$, 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.