On the tractability of defensive alliance problem
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| Titel: | On the tractability of defensive alliance problem |
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| Autoren: | Sangam Balchandar Reddy, Anjeneya Swami Kare |
| Quelle: | Discrete Applied Mathematics. 380:116-127 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Elsevier BV, 2026. |
| Publikationsjahr: | 2026 |
| Schlagwörter: | FOS: Computer and information sciences, Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms, Data Structures and Algorithms (cs.DS), Computational Complexity (cs.CC) |
| Beschreibung: | Given a graph $G = (V, E)$, a non-empty set $S \subseteq V$ is a defensive alliance, if for every vertex $v \in S$, the majority of its closed neighbours are in $S$, that is, $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$. The decision version of the problem is known to be NP-Complete even when restricted to split and bipartite graphs. The problem is \textit{fixed-parameter tractable} for the parameters solution size, vertex cover number and neighbourhood diversity. For the parameters treewidth and feedback vertex set number, the problem is W[1]-hard. \\ \hspace*{2em} In this paper, we study the defensive alliance problem for graphs with bounded degree. We show that the problem is \textit{polynomial-time solvable} on graphs with maximum degree at most 5 and NP-Complete on graphs with maximum degree 6. This rules out the fixed-parameter tractability of the problem for the parameter maximum degree of the graph. We also consider the problem from the standpoint of parameterized complexity. We provide an FPT algorithm using the Integer Linear Programming approach for the parameter distance to clique. We also answer an open question posed in \cite{AG2} by providing an FPT algorithm for the parameter twin cover. 20 pages |
| Publikationsart: | Article |
| Sprache: | English |
| ISSN: | 0166-218X |
| DOI: | 10.1016/j.dam.2025.09.008 |
| DOI: | 10.48550/arxiv.2307.09760 |
| Zugangs-URL: | http://arxiv.org/abs/2307.09760 |
| Rights: | Elsevier TDM CC BY |
| Dokumentencode: | edsair.doi.dedup.....da4a9d839a2cd1d45d4e1806d99ed9d2 |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Given a graph $G = (V, E)$, a non-empty set $S \subseteq V$ is a defensive alliance, if for every vertex $v \in S$, the majority of its closed neighbours are in $S$, that is, $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$. The decision version of the problem is known to be NP-Complete even when restricted to split and bipartite graphs. The problem is \textit{fixed-parameter tractable} for the parameters solution size, vertex cover number and neighbourhood diversity. For the parameters treewidth and feedback vertex set number, the problem is W[1]-hard. \\ \hspace*{2em} In this paper, we study the defensive alliance problem for graphs with bounded degree. We show that the problem is \textit{polynomial-time solvable} on graphs with maximum degree at most 5 and NP-Complete on graphs with maximum degree 6. This rules out the fixed-parameter tractability of the problem for the parameter maximum degree of the graph. We also consider the problem from the standpoint of parameterized complexity. We provide an FPT algorithm using the Integer Linear Programming approach for the parameter distance to clique. We also answer an open question posed in \cite{AG2} by providing an FPT algorithm for the parameter twin cover.<br />20 pages |
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| ISSN: | 0166218X |
| DOI: | 10.1016/j.dam.2025.09.008 |