On the complexity of the minimum outer-connected dominating set problem in graphs

Abstract

For a graph $G=(V,E)$, a dominating set is a set $D\subseteq V$ such that every vertex $v\in V\setminus D$ has a neighbor in $D$. The minimum outer-connected dominating set (Min-Outer-Connected-Dom-Set) problem for a graph $G$ is to find a dominating set $D$ of $G$ such that $G[V\setminus D]$, the induced subgraph by $G$ on $V\setminus D$, is connected and the cardinality of $D$ is minimized. In this paper, we consider the complexity of the Min-Outer-Connected-Dom-Set problem. In particular, we show that the decision version of the Min-Outer-Connected-Dom-Set problem is NP-complete for split graphs, a well known subclass of chordal graphs. We also consider the approximability of the Min-Outer-Connected-Dom-Set problem. We show that the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of $(1-\varepsilon ) \ln |V|$ for any $\varepsilon >0$, unless NP $\subseteq$ DTIME($|V|^{\log \log |V|}$). For sufficiently large values of $\varDelta$, we show that for graphs with maximum degree $\varDelta$, the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of $\ln \varDelta -C \ln \ln \varDelta$ for some constant $C$, unless P $=$ NP. On the positive side, we present a $\ln (\varDelta +1)+1$-factor approximation algorithm for the Min-Outer-Connected-Dom-Set problem for general graphs. We show that the Min-Outer-Connected-Dom-Set problem is APX-complete for graphs of maximum degree 4.