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.

Source : oai:arXiv.org:1612.08234

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

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