Graphs with equal Grundy domination and independence number

Abstract

The Grundy domination number, γgr(G), of a graph G is the maximum length of a sequence (v1,v2,…,vk) of vertices in G such that for every i∈{2,…,k}, the closed neighborhood N[vi] contains a vertex that does not belong to any closed neighborhood N[vj], where j<i. It is well known that the Grundy domination number of any graph G is greater than or equal to the upper domination number Γ(G), which is in turn greater than or equal to the independence number α(G). In this paper, we initiate the study of the class of graphs G with Γ(G)=γgr(G) and its subclass consisting of graphs G with α(G)=γgr(G). We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs G with Γ(G)=γgr(G) and present large families of such graphs.