### Connected k-forcing sets of graphs and splitting graphs

#### Abstract

The notion of k-forcing number of a graph was introduced by Amos et al. For a given graph G and a given subset I of the vertices of the graph G, the vertices in I are known as initially colored black vertices and the vertices in V(G)−I are known as not initially colored black vertices or white vertices. The set I is a k-forcing set of a graph G if all vertices in G eventually colored black after applying the following color changing rule: If a black colored vertex is adjacent to at most k-white vertices, then the white vertices change to be colored black. The cardinality of a smallest k-forcing set is known as the k-forcing number Zk(G) of the graph G. If the sub graph induced by the vertices in I are connected, then I is called the connected k-forcing set. The minimum cardinality of such a set is called the connected k-forcing number of G and is denoted by Zck(G). This manuscript is intended to study the connected k-forcing number of graphs and the splitting graphs.

**Published:**2020-03-06

**How to Cite this Article:**K.P. Premodkmuar, Charles Dominic, Baby Chacko, Connected k-forcing sets of graphs and splitting graphs, J. Math. Comput. Sci., 10 (2020), 656-680 Copyright © 2020 K.P. Premodkmuar, 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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