Algorithmic aspects of open neighborhood location-domination in graphs

A set D⊆V of a graph G=(V,E) is called an open neighborhood locating-dominating set (OLD-set) if (i) NG(v)∩D≠0̸ for all v∈V, and (ii) NG(u)∩D≠NG(v)∩D for every pair of distinct vertices u,v∈V. Given a graph G=(V,E), the Min OLD-set problem is to find an OLD-set of minimum cardinality. Given a graph G=(V,E) and a positive integer k, the Decide OLD-set problem is to decide whether G has an OLD-set of cardinality at most k. The Decide OLD-set problem is known to NP-complete for general graphs. In this paper we extend the NP-completeness result of the Decide OLD-set problem by showing that it remains NP-complete for bipartite graphs, planar graphs, split graphs and doubly chordal graphs. We prove that the Decide OLD-set problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for the Min OLD-set problem in trees. We also propose a (2+3lnΔ)-approximation algorithm for the Min OLD-set problem and show that the Min OLD-set problem cannot be approximated within 12(1−ϵ)ln|V| for any ϵ>0 unless NP⊆DTIME(|V|O(loglog|V|)). Finally, we prove that the Min OLD-set problem is APX-complete for bipartite graphs of maximum degree 3.