The Nonsplit Resolving Domination Polynomial of a Graph

Abstract

Metric representation of a vertex v in a graph G with an ordered subset R = {a1, a2, ... , ak} of vertices of G is the kvector r(v|R) = (d(v, a1), d(v, a2), ... , d(v, ak)), where d(v, a) is the distance between v and a in G. The set R is called a Resolving set of G , if any two distinct vertices of G have distinct representation with respect to R . The cardinality of a minimum resolving in G is called a dimension of G, and is denoted by dim(G). In a graph G = (V, E), A subset D ⊆ V is a nonsplit resolving dominating set of G if it is a resolving, and nonsplit dominating set of G. The minimum cardinality of a nonsplit resolving dominating set of Gis known as a nonsplit resolving domination number of G, and is represented by γnsr(G) . In network reliability domination polynomial has found its application [20], a resolving set has diverse applications which includes verification of network and its discovery, mastermind game, robot navigation, problems of pattern recognition, image processing, optimization and combinatorial search [19]. Here, we are introducing nonsplit resolving domination polynomial of G. Some properties of the nonsplit Resolving domination polynomial of Gare studied and nonsplit resolving domination polynomials of some well-known families of graphs are calculated.