## Rajaati, M. and Hooshmandasl, M. R. and Dinneen, M. J. and Shakiba, A. - On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

dmtcs:2615 - Discrete Mathematics & Theoretical Computer Science, July 31, 2018, vol. 20 no. 2 - https://doi.org/10.23638/DMTCS-20-2-2
On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

Authors: Rajaati, M. and Hooshmandasl, M. R. and Dinneen, M. J. and Shakiba, A.

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.

Volume: vol. 20 no. 2
Section: Graph Theory
Published on: July 31, 2018
Submitted on: December 28, 2016
Keywords: Computer Science - Discrete Mathematics,Computer Science - Data Structures and Algorithms,05C85,G.2.2