k-restricted edge connectivity in (p+1)-clique-free graphs

Let G be a graph with vertex set V(G) and edge set E(G). An edge subset S c E(G) is called a k-restricted edge cut if G - S is not connected and every component of G - S has at least k vertices. The k-restricted edge connectivity of a connected graph G, denoted by 4(G), is defined as the cardinality of a minimum k-restricted edge cut. Let [X, (X) over bar] denote the set of edges between a vertex set X subset of V (G) and its complement (X) over bar = V(G)\X. A vertex set X subset of V (G) is called a lambda(k)-fragment if [X, (X) over bar] is a minimum k-restricted edge cut of G. Let xi(k)(G) = min{vertical bar[X, (X) over bar]vertical bar : vertical bar X vertical bar = k, G[X] is connected}. In this work, we give a lower bound on the cardinality of lambda(k)-fragments of a graph G satisfying lambda(k)(G) < xi(k)(G) and containing no (p + 1)-cliques. As a consequence of this result, we show a sufficient condition for a graph G with lambda(k)(G) = xi(k)(G). (C) 2014 Elsevier B.V. All rights reserved.