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 orincident 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 setsof $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ iscalled a minimum mixed dominating set. The mixed domination set (MDS) problemis to find a minimum mixed dominating set for a graph $G$ and is known to be anNP-complete problem. In this paper, we present a novel approach to find all ofthe mixed dominating sets, called the AMDS problem, of a graph with boundedtree-width $tw$. Our new technique of assigning power values to edges andvertices, and combining with dynamic programming, leads to a fixed-parameteralgorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS isfixed-parameter tractable with respect to tree-width. In addition, wetheoretically 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 algorithm