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.