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

Abstract

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 $O(3^{t{w}^2}\times t{w}^2\times |V|)$. 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.

Accepted for the publication in the Journal of Discrete Mathematics & Theoretical Computer Science (DMTCS). 25 pages, 4 figures, 17 tables, 4 algorithms

References in corpus (1)

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

Cited by in corpus (2)

• On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width
• New Algorithms for Mixed Dominating Set