Algorithms and Hardness for Signed Domination

Abstract

A signed dominating function for a graph $G=(V, E)$ is a function $f$: $V \rightarrow \{ +1, -1\}$ such that for all $v \in V$, the sum of the function values over the closed neighborhood of $v$ is at least one. The weight $w(f(V))$ of signed dominating function $f$ for vertex set $V$ is the sum of $f(v)$ for $v \in V$. The signed domination number $\gamma _s$ of $G$ is the minimum weight of a signed dominating function for $G$. The signed domination (SD) problem asks for a signed dominating function which contributes the signed domination number. First we show that the SD problem is W[2]-hard. Next we show that the SD problem on graphs of maximum degree six is APX-hard. Then we present constant-factor approximation algorithms for the SD problem on subcubic graphs, graphs of maximum degree four, and graphs of maximum degree five, respectively. In addition, we present an alternative and more direct proof for the NP-completeness of the SD problem on subcubic planar bipartite graphs. Lastly, we obtain an $O^{*}(5.1957^k)$-time FPT-algorithm for the SD problem on subcubic graphs $G$, where $k$ is the signed domination number of $G$.