On the first-order edge tenacity of a graph

The first-order edge-tenacity T-1 (G) of a graph G is defined as T-1(G) = min {vertical bar X vertical bar + tau(G - X)/omega(G - X) - 1} where the minimum is taken over every edge-cutset X that separates G into omega(G - X) components, and by tau(G - X) we denote the order (the number of edges) of a largest component of G - X. The objective of this paper is to study this concept of edge-tenacity and determining this quantity for some special classes of graphs. We calculate the first-order edge-tenacity of a complete n-partite graph. We shall obtain the first-order edge-tenacity of maximal planar graphs, maximal outerplanar graphs, and k-trees. Let G be a graph of order p and size q, we shall call the least integer r,1 <= r <= p - 1, with T-r(G) = q/p-r the balancity of G and denote it by b (G). Note that the balancity exists since T-r(G) = q/p-r if r = p - 1. In general, it is difficult to determine the balancity of a graph. In this paper, we shall first determine the balancity of a special class of graphs and use this to find an upper bound for the balancity of an arbitrary graph. (C) 2015 Elsevier B.V. All rights reserved.