On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width
A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominat...
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| Vydáno v: | Discrete mathematics and theoretical computer science Ročník 20 no. 2; číslo Graph Theory |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Discrete Mathematics & Theoretical Computer Science
31.07.2018
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| Témata: | |
| ISSN: | 1365-8050, 1365-8050 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} \times |V|)$ time.
Comment: Accepted for the publication in the Journal of Discrete Mathematics & Theoretical Computer Science (DMTCS). 25 pages, 4 figures, 17 tables, 4 algorithms |
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| ISSN: | 1365-8050 1365-8050 |
| DOI: | 10.23638/DMTCS-20-2-2 |