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$k$-secure sets and $k$-security number of a graph | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 1، دوره 10، شماره 2، شهریور 2025، صفحه 245-255 اصل مقاله (427.36 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28048.1427 | ||
| نویسندگان | ||
| K. Karthik؛ Chandru Hegde* | ||
| Department of Mathematics, Mangalore University, Mangalagangothri, Mangalore, India | ||
| چکیده | ||
| Let $G=(V, E)$ be a simple connected graph. A nonempty set $S\subseteq V$ is a secure set if every attack on $S$ is defendable. In this paper, $k$-secure sets are introduced as a generalization of secure sets. For any integer $k\geq 0$, a nonempty subset $S$ of $V$ is a $k$-secure set if, for each attack on $S$, there is a defense of $S$ such that for every $v\in S$, the defending set of $v$ contains at least $k$ more elements than that of the attacking set of $v$, whenever the vertex $v$ has neighbors outside $S$. The cardinality of a minimum $k$-secure set in $G$ is the $k$-security number of $G$. Some properties of $k$-secure sets are discussed and a characterization of $k$-secure sets is obtained. Also, 1-security numbers of certain classes of graphs are determined. | ||
| کلیدواژه‌ها | ||
| Secure sets؛ Alliances؛ Security number؛ k-Secure Sets | ||
| مراجع | ||
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