Applying the (1,2)-pitchfork domination and it's inverse on some special graphs

Abstract

Let G be a finite simple and undirected graph without isolated vertices. For any non-negative integers j and k, a subset D of V is called a pitchfork dominating set if every vertex in D dominates at least j and at most k vertices of V - D. A subset D -1 of V - D is an inverse pitchfork dominating set if it is a dominating set. The pitchfork domination number of G, denoted by  pf (G) is a minimum cardinality over all pitchfork dominating sets in G. The inverse pitchfork domination number of G, denoted by pf-1 (G) is a minimum cardinality over all inverse pitchfork dominating sets in G. In this paper, pitchfork dominations and it's inverse are applied when j = 1 and k = 2 on some standard graphs such as: tadpole graph, lollipop graph, lollipop flower graph , daisy graph and Barbell graph.