Open Neighborhood Locating-Dominating Set in Graphs: Complexity and Algorithms

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 ≠ empty set for all v ≠ V, and (ii) NG (u)n D≠ N_G (v)n 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. The cardinality of a minimum OLD-set of G is called the open neighborhood location-domination number of G, and is denoted by γ_old (G). Given a graph G 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 be NP-complete for bipartite graphs. In this paper, we strengthen this NP-complete result by showing that the Decide OLD-set problem remains NP-complete for perfect elimination bipartite graphs, a subclass of bipartite graphs. Then, we show that the Min OLD-set problem can be solved in polynomial time in chain graphs, a subclass of perfect elimination bipartite graphs. We show that for a graph G, γ_old (G)=/2n Δ(G) + 2, where n denotes the number of vertices in G, and Δ (G) denotes the maximum degree of G. As a consequence we obtain a Δ (G) + 2 2-approximation algorithm for the Min OLD-set problem. Finally, we prove that the Min OLD-set problem is APX-complete for chordal graphs with maximum degree 4.