### 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.

**Published:**2020-08-27

**How to Cite this Article:**K.P. Premodkumar, Charles Dominic, Baby Chacko, Two distance forcing number of a graph, J. Math. Comput. Sci., 10 (2020), 2233-2248 Copyright © 2020 K.P. Premodkumar, Charles Dominic, Baby Chacko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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