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

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V , |NG[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 Outerconnected 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 show that the Outer-connected Domination Decision problem can be solved in linear-time for graphs of bounded tree-width. We propose a ∆(G)approximation algorithm for the Minimum Outer-connected Domination problem, where ∆(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 | log |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. Submitted: March 2014 Reviewed: June 2014 Revised: July 2014 Accepted: July 2014 Final: December 2014 Published: December 2014 Article type: Regular paper Communicated by: S. Pal and K. Sadakane A preliminary version of this work appeared in the proceedings of WALCOM 2014 [16]. E-mail addresses: [email protected] (B.S. Panda) [email protected] (Arti Pandey) 494 B.S. Panda, Arti Pandey Outer-connected Dominating Set in Graphs