Stabile and Critical Pitchfork Domination in Graphs

Abstract

Let G be a nite, simple, undirected graph and without isolated vertices. A subset D of V is a pitchfork dominating set if j jN(v) \ (V 􀀀 D)j k for every v 2 D
and any j and k integers. In this paper, The eects of adding or removing an edge and removing vertex from the graph are studied on the pitchfork domination number pf (G) for j = 1 and k = 2. Some graphs didn't eects on this changing, while pf (G) of other graphs were increasing or decreasing. The study of these eects has an important advantages to learn the ways of treatments to any added or damaged of any nods (vertices) or links (edges) of the system or networks to avoid losing some properties of the system and to give the best services with minimum costs.