On Extremal Graphs for Zero Forcing Number

Liang, Yi-Ping; Li, Jianxi; Xu, Shou‐Jun

doi:10.1007/s00373-022-02591-y

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2023

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“…Caro and Pepper [4] used a greedy algorithm to obtain an improved bound , where is the minimum degree of G . We gave a complete characterisation of the extremal graphs for this bound in [10]. For the total forcing number, Davila and Henning [7] showed that if G is a connected graph of order with maximum degree , then , with equality if and only if or .…”

Section: Introductionmentioning

confidence: 99%

Upper Bounds on the Semitotal Forcing Number of Graphs

Liang

Chen²,

Xu³

2023

Bull. Aust. Math. Soc.

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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 $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$ , then $F_{t2}(G)\leq (\Delta-1)n/\Delta$ , with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$ .

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