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 t 2 ( G ) is the minimum cardinality of a semitotal forcing set in G . In this paper, we prove that if G (cid:54) = K 4 is a connected claw-free cubic graph of order n , then F t 2 ( G ) ≤ 38 n + 1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.