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Azami Aghdash, Akbar, Jafari Rad, Nader, Vakili, Bahram. (1404). Stability with respect to total restrained domination in bipartite graphs. فناوری آموزش, 10(3), 263-272. doi: 10.22061/jdma.2025.11697.1113
Akbar Azami Aghdash; Nader Jafari Rad; Bahram Vakili. "Stability with respect to total restrained domination in bipartite graphs". فناوری آموزش, 10, 3, 1404, 263-272. doi: 10.22061/jdma.2025.11697.1113
Azami Aghdash, Akbar, Jafari Rad, Nader, Vakili, Bahram. (1404). 'Stability with respect to total restrained domination in bipartite graphs', فناوری آموزش, 10(3), pp. 263-272. doi: 10.22061/jdma.2025.11697.1113
Azami Aghdash, Akbar, Jafari Rad, Nader, Vakili, Bahram. Stability with respect to total restrained domination in bipartite graphs. فناوری آموزش, 1404; 10(3): 263-272. doi: 10.22061/jdma.2025.11697.1113

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
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