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

Part of: Graph theory

Published online by Cambridge University Press:  12 February 2024

JIE CHEN Open the ORCID record for JIE CHEN [Opens in a new window] ,
HONGZHANG CHEN Open the ORCID record for HONGZHANG CHEN [Opens in a new window]  and
SHOU-JUN XU Open the ORCID record for SHOU-JUN XU [Opens in a new window]
JIE CHEN
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: ChenJieJie2023@hotmail.com
HONGZHANG CHEN
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: mnhzchern@gmail.com
SHOU-JUN XU*
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China
*
e-mail: shjxu@lzu.edu.cn
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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].

Keywords

claw-free graphedge weighting functionsemitotal domination

MSC classification

Primary: 05C69: Dominating sets, independent sets, cliques
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was funded in part by the National Natural Science Foundation of China (Grant No. 12071194) and the Chongqing Natural Science Foundation Innovation and Development Joint Fund (Municipal Education Commission) (Grant No. CSTB2022NSCQ-LZX0003).

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