Zero forcing number of degree splitting graphs and complete degree splitting graphs

A subset Z subset of V(G) of initially colored black vertices of a graph G is known as a zero forcing set if we can alter the color of all vertices in G as black by iteratively applying the subsequent color change condition. At each step, any black colored vertex has exactly one white neighbor, then change the color of this white vertex as black. The zero forcing number Z(G), is the minimum number of vertices in a zero forcing set Z of G (see [11]). In this paper, we compute the zero forcing number of the degree splitting graph (DS-Graph) and the complete degree splitting graph (CDS-Graph) of a graph. We prove that for any simple graph, Z[DS(G)] <= k + t, where Z(G) = k and t is the number of newly introduced vertices in DS(G) to construct it.