Secure equitable subdivision number of graphs

Abstract

A dominating set S of a graph G(V, E) is called a secure dominating set, if for each vertex x belongs to V(G) \ S there exists at least one vertex y in S such that y  belongs to the neighbourhood of x and a vertex y can be replaced by a vertex x without losing domination property. A secure dominating set S is a secure equitable dominating set if for each x in V(G) \ S and the vertex y in S, which replaces x, we have |deg(x)-deg(y)| is less than or equal to 1. In this paper, we initiate the study of a new domination parameter, the secure equitable subdivision number of graphs. Also, we introduce the definitions of the secure equitable subdivision critical graphs and the secure equitable subdivision stable graphs. Further, we study the construction of a secure equitable subdivision critical graph.