Doubly connected pitchfork domination and it’s inverse in graphs

Abstract

Let G be a finite, simple, undirected graph and without isolated vertices. A subset D of V is a pitchfork dominating set if every vertex v ? D dominates at least j and at most k vertices of V ? D, for any j and k integers. A subset D?¹ of V ? D is an inverse pitchfork dominating set if D?¹ is a dominating set. The domination number of G, denoted by ?pf (G) is a minimum cardinality over all pitchfork dominating sets in G. The inverse domination number of G, denoted by ??¹ pf (G) is a minimum cardinality over all inverse pitchfork dominating sets in G. In this paper, a special modified pitchfork dominations called doubly connected pitchfork domination and it’s inverse are introduced when j = 1 and k = 2. Some properties and bounds are studied with respect to the order and the size of the graph. These modified dominations are applied and evaluated for several well known and complement graphs.