On the Semitotal Forcing Number of a Graph

Semitotal forcing in claw-free cubic graphs

“…By Lemma 4, we note that if G is a connected claw-free cubic graph with no triangle-units, then G ∈ N cubic . The semitotal forcing number of a diamond-necklace was determined by Chen in [7].…”

“…In this paper, we focus on the semitotal forcing, which was first introduced by Chen in [7]. A set S of vertices in G is a semitotal forcing set of G if it is a forcing set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number, denoted by F t2 (G), is the cardinality of a minimum semitotal forcing set of G.…”

“…Since every total forcing set is also a semitotal forcing set, and since every semitotal forcing set is a forcing set, we have the following chain of inequalities [7]. For every isolate-free graph G, F (G) ≤ F t2 (G) ≤ F t (G).…”

“…Let G = K 4 be a connected claw-free cubic graph of order n. Davila and Henning [8] showed that F t (G) ≤ 1 2 n, with equality if and only if G ∈ N cubic or G is the prism C 3 K 2 ; and then these two authors [11] showed that F (G) ≤ 1 3 n + 1 unless G = N 2 . Chen [7] showed that F t2 (G) ≤ 1 2 n, with equality if and only if G is the diamond-necklace N 2 or the prism C 3 K 2 . In this paper, we improve this upper bound on the semitotal forcing number of G: F t2 (G) ≤ 3 8 n + 1, and the graphs achieving equality in this bound are characterized i.e., Theorem 13.…”

Upper Bounds on the Semitotal Forcing Number of Graphs

“…The (zero) forcing number of a graph was first introduced by the AIM Minimum Rank-Special Graphs Work Group [2] to bound the maximum nullity/minimum rank of the family of symmetric matrices associated with a graph. Total forcing and semitotal forcing are two variations of forcing, which were first introduced and studied by Davila and Kenter [8] and Chen [6]. The definitions are as follows.…”