Two distance forcing number of a graph

Abstract

Motivated from the graph parameters namely zero forcing number, k-forcing number and the connected k-forcing number, in this article, we introduce a new parameter known as the 2-distance forcing number. Assume that each vertex of a graph G = (V(G),E(G)) is colored as either white or black. Consider the set Z2d of black colored vertices of the graph G. The color change rule changes the color of a white vertex v to black if the white vertex v is the only 2-distance white neighbor of a black vertex u. The set Z2d is called a two distance forcing set of G if all vertices of the graph G will be turned black after limited applications of the color change rule. The 2-distance forcing number of G, denoted by Z2d(G), is the minimum of |Z2d| over all 2- distance forcing sets Z2d ⊆ V(G). This manuscript is intended to study the 2-distance forcing number of some graphs. We find the exact value of the 2-distance forcing number of graphs such as the pineapple graph, gear graph, jelly fish graph, helm graph, sunflower graph, comet graph and the n-pan graph.