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Original Articles

On the edge-tenacity of the middle graph of a graph

Pages 551-558 | Received 06 Jul 2004, Published online: 20 Aug 2006
 

Abstract

We consider the problem of efficiently breaking a graph into small components by removing edges. One measure of how easily this can be done is the edge-tenacity. Given a set of edges of G, the score of S is defined as sc(S)=[| S|+τ (GS)]/[w(GS)]. Formally, the edge-tenacity of a graph G is defined as T′(G)=min sc(S), where the minimum is taken over all edge-sets S of G. A subset S of E(G) is said to be a T′-set of G if T′(G)=sc(S). Note that if G is disconnected, the set S may be empty. For any graph G, τ(GS) is the number of vertices in the largest component of GS and w(GS) is the number of components of GS. The middle graph M(G) of a graph G is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we give the edge-tenacity of the middle graph of specific families of graphs and its relationships with other parameters.

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