Semitotal forcing in claw-free cubic graphs

Abstract

For an isolate-free graph G = (V (G), E(G)), a set S ⊆ V (G) is called a semitotal forcing set of G if it is a forcing set (or a zero forcing set) of G and 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 if G = K 4 is a connected claw-free cubic graph of order n, then F t2 (G) ≤ 3 8 n + 1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.