# On the Arrangement of Cliques in Chordal Graphs with respect to the Cuts

Chandran, Sunil L and Narayanaswamy, NS (2009) On the Arrangement of Cliques in Chordal Graphs with respect to the Cuts. In: 10th International Computing and Combinatorics Conference, AUG 17-20, 2004, Cheju Isl, SOUTH KOREA, pp. 11-19.

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## Abstract

A cut (A, B) (where B = V - A) in a graph G = (V, E) 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 is an element of 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 kappa(G) + vertices (where kappa(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 &lt;= i &lt;= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(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 Omega(k(2)), where kappa(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 kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.

Item Type: Conference Paper Ars Combinatoria Springer Copyright of this article belongs to Springer. Division of Electrical Sciences > Computer Science & Automation 06 Jan 2010 05:38 28 Feb 2019 08:36 http://eprints.iisc.ac.in/id/eprint/21484