Algorithm and Hardness Results for Outer-connected Dominating Set in Graphs

Abstract

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V, |N _G [v] ∩ D| ≥ 1, and the induced subgraph of G on V ∖ D is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V,E), the Outer-connected Domination Decision problem is to decide whether G has an outer-connected dominating set of cardinality at most k. The Outer-connected Domination Decision problem is known to be NP-complete for bipartite graphs. In this paper, we strengthen this NP-completeness result by showing that the Outer-connected Domination Decision problem remains NP-complete for perfect elimination bipartite graphs. On the positive side, we propose a linear time algorithm for computing a minimum outer-connected dominating set of a chain graph, a subclass of bipartite graphs. We propose a $\varDelta(G)$-approximation algorithm for the Minimum Outer-connected Domination problem, where $\varDelta(G)$ is the maximum degree of G. On the negative side, we prove that the Minimum Outer-connected Domination problem cannot be approximated within a factor of (1 − ε)ln |V| for any ε > 0, unless NP ⊆ DTIME(|V|^O(loglog|V|)). We also show that the Minimum Outer-connected Domination problem is APX-complete for graphs with bounded degree 4 and for bipartite graphs with bounded degree 7.