On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

• Published in: Discrete mathematics and theoretical computer science Vol. 20 no. 2; no. Graph Theory
• Main Authors: Rajaati, M., Hooshmandasl, M. R., Dinneen, M. J., Shakiba, A.
• Format: Journal Article
• Language: English
• Published: Discrete Mathematics & Theoretical Computer Science 31.07.2018
• Subjects: 05c85; computer science - data structures and algorithms; computer science - discrete mathematics; g.2.2
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.23638/DMTCS-20-2-2

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