Characterization of Some Claw-Free Graphs in Co-Secure Domination Number

Yuexin Zhang, Jiayuan Zhang, Siwen Jing, Xiaodong Chen () and Liming Xiong
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Yuexin Zhang: School of Mathematics, Liaoning Normal University, Dalian 116029, China
Jiayuan Zhang: School of Mathematics, Liaoning Normal University, Dalian 116029, China
Siwen Jing: School of Mathematics, Liaoning Normal University, Dalian 116029, China
Xiaodong Chen: School of Mathematics, Liaoning Normal University, Dalian 116029, China
Liming Xiong: School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

Mathematics, 2025, vol. 13, issue 15, 1-12

Abstract: For a vertex subset S of a graph G , if each vertex of G is either in S or adjacent to some vertex in S , then S is a dominating set of G . Let S be a dominating set of a graph G . If each vertex v not in S has a neighbor u in S such that ( S \ { u } ) ∪ { v } is also a dominating set of G , then S is a secure dominating set of G . If each vertex u in S has a neighbor v not in S such that ( S \ { u } ) ∪ { v } is also a dominating set of G , then S is a co-secure dominating set of G . The minimum cardinality of a secure (resp. co-secure) dominating set of G is the secure (resp. co-secure) domination number of G . Arumugam et al. proposed the questions to characterize a graph G such that the co-secure domination number of G equals the independence number and the secure domination number of G , respectively. Inspired by those questions, in this paper, we obtain two classes of claw-free graphs such that the co-secure domination number equal the independence number and the secure domination number. Our results provide some theoretical basis of claw-free graphs for networks.

Keywords: co-secure dominating set; secure dominating set; co-secure domination number; secure domination number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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