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

Abstract

For a graph $$G=(V,E)$$G=(V,E), a dominating set is a set $$D\subseteq V$$D⊆V such that every vertex $$v\in V\setminus D$$v?V\D has a neighbor in $$D$$D. The minimum outer-connected dominating set (Min-Outer-Connected-Dom-Set) problem for a graph $$G$$G is to find a dominating set $$D$$D of $$G$$G such that $$G[V\setminus D]$$G[V\D], the induced subgraph by $$G$$G on $$V\setminus D$$V\D, is connected and the cardinality of $$D$$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 $$(1-\varepsilon ) \ln |V|$$(1-?)ln|V| for any $$\varepsilon >0$$?>0, unless NP $$\subseteq $$⊆ DTIME($$|V|^{\log \log |V|}$$|V|loglog|V|). For sufficiently large values of $$\varDelta $$Δ, we show that for graphs with maximum degree $$\varDelta $$Δ, the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of $$\ln \varDelta -C \ln \ln \varDelta $$lnΔ-ClnlnΔ for some constant $$C$$C, unless P $$=$$= NP. On the positive side, we present a $$\ln (\varDelta +1)+1$$ln(Δ+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.