Dominator Coloring Number of Middle Graph

Abstract

Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. In this paper, as an extension of Dominator coloring some standard results for the middle graph of path and cycle has been discussed. Let G = (V(G), E(G)) be a graph with n = |V(G)| and m = |E(G)|. For any vertex v V(G), the open neighborhood of v is the set N (v) = {u| uv  E (G)} and the closed neighborhood is the set N(v) = N(v)  v. Similarly, for any set SV (G), N(S) =  vs N (v) -S and N(S) = N(S)  S. A set S is a dominating set if N(S)