UPPER BOUNDS ON THE SEMITOTAL FORCING NUMBER OF GRAPHS

Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number F-t2(G) is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if G ? K(n )is a connected graph of order n = 4 with maximum degree ? = 2, then F-t2(G) = (? - 1)n/?, with equality if and only if either G = C(4)or G = P(4)or G = K-?,K-?.