EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS

IF 0.6 4区数学 Q3 MATHEMATICS Bulletin of the Australian Mathematical Society Pub Date : 2024-02-12 DOI:10.1017/s0004972724000017

JIE CHEN, HONGZHANG CHEN, SHOU-JUN XU

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Abstract

In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by

$\gamma _{t2}(G)$

, is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if

$G\neq N_2$

is a connected claw-free graph of order

$n\geq 6$

with minimum degree

$\delta (G)\geq 3$

, then

$\gamma _{t2}(G)\leq \frac{4}{11}n$

and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin.33(5) (2017), 1119–1130].

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无爪图半总支配集上的边加权函数

在无孤立图 G 中，如果顶点子集 S 是 G 的支配集，且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内，则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 $\gamma _{t2}(G)$ 表示，是 G 中半总支配集的最小心数。利用半总支配集上的边加权函数，我们证明了如果 $Gneq N_2$ 是一个阶数为 $n/geq 6$ 且最小度数为 $\delta (G)\geq 3$ 的无连接爪图，那么 $gamma _{t2}(G)\leq \frac{4}{11}n$ 并且这个约束是尖锐的，推翻了 Zhu 等人提出的猜想。['无爪立方图中的半总支配'，Graphs Combin.33(5) (2017), 1119-1130].

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Bulletin of the Australian Mathematical Society 数学-数学

CiteScore

1.20

自引率

14.30%

发文量

149

审稿时长

4-8 weeks

期刊介绍： Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way. Published Bi-monthly Published for the Australian Mathematical Society

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