Differential in infrastructure networks

Abstract

Let G = (V, E) be a graph of order n and let B(D) be the set of vertices in V \ D that have a neighbor in the vertex set D. The differential of a vertex set D is defined as ∂(D) = |B(D)| − |D| and the maximum value of ∂(D) for any subset D of V is the differential of G. A set D of vertices of a graph G is said to be a dominating set if every vertex in V \ D is adjacent to a vertex in D. G is a dominant differential graph if it contains a ∂-set which is also a dominating set. This paper is devoted to the computation of differential of wheel, cycle and path-related graphs as infrastructure networks. Furthermore, dominant differential wheel, cycle and path-related types of networks are recognized.