Pitchfork domination in graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let G = (V, E) be a finite, simple and undirected graph without isolated vertices, a subset D of V is a pitchfork dominating set if every vertex v is an element of D dominates at least j and at most k vertices of V - D, where j and k are non-negative integers. The domination number of G, denotes gamma(pf)(G) is a minimum cardinality over all pitchfork dominating sets in G. In this work, pitchfork domination when j = 1 and k = 2 is studied. Some bounds on gamma(pf)(G) related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph G without isolated vertices have a pitchfork domination or not?