Stability with respect to total restrained domination in bipartite graphs

Document Type : Full Length Article

Authors

1 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, I. R. Iran

2 Department of Mathematics, Shahed University, Tehran, I. R. Iran

Abstract

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 γtr(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 stγtr(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.

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References

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Journal of Discrete Mathematics and Its Applications
Volume 10, Issue 3
September 2025
Pages 263-272

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History

  • Receive Date: 28 January 2025
  • Revise Date: 03 March 2025
  • Accept Date: 08 March 2025
  • First Publish Date: 08 July 2025
  • Publish Date: 01 September 2025

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