Some results on the exact 1-step domination graphs

Abstract

An exact 1-step dominating set in a graph G is a subset S of vertices of G such that ( ) 1 N v S   for every vertex ( ) v V G  . A graph is an exact 1-step domination graph if it contains an exact 1-step dominating set. In this paper, we obtain new upper bounds on the size of exact 1-step domination graphs. We also present an upper bound on the total domination number of an exact 1-step domination tree and characterize trees achieving equality for this bound.