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Stability with respect to total restrained domination in bipartite graphs | ||
| Journal of Discrete Mathematics and Its Applications | ||
| دوره 10، شماره 3، آذر 2025، صفحه 263-272 اصل مقاله (279.06 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2025.11697.1113 | ||
| نویسندگان | ||
| Akbar Azami Aghdash1؛ Nader Jafari Rad* 2؛ Bahram Vakili1 | ||
| 1Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, I. R. Iran | ||
| 2Department of Mathematics, Shahed University, Tehran, I. R. Iran | ||
| تاریخ دریافت: 09 بهمن 1403، تاریخ بازنگری: 13 اسفند 1403، تاریخ پذیرش: 18 اسفند 1403 | ||
| چکیده | ||
| In a graph $G = (V, E)$ with no isolated vertices, a subset $D$ of vertices is said to be a total dominating set (abbreviated TDS) if it has the property that every vertex of $G$ is adjacent to some vertex in $D$. A TDS $D$ is said to be a total restrained dominating set (abbreviated TRDS) if it has a further property that any vertex in $V-D$ is also adjacent to a vertex in $V-D$. Given the isolate-free graph $G$, the total restrained domination number of $G$, which we denote it by $\gamma_{tr}(G)$, is the minimum cardinality of a TRDS of $G$. The minimum number of vertices of the graph $G$ whose removal changes the total restrained domination number of $G$ is called the total restrained domination stability number of $G$, and is denoted by $st_{\gamma_{tr}}(G)$. In this paper we study this variant in bipartite graphs. We show that the related decision problem related to this variant is NP-hard in bipartite graphs. We also determine the total restrained stability number in some families of graphs, including the families of trees and unicyclic graphs. | ||
| کلیدواژهها | ||
| total dominating set؛ total restrained dominating set؛ bipartite graph | ||
| مراجع | ||
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