Algorithmic Aspects of Disjunctive Domination in Graphs

For a graph G = (V,E), a set D ⊆ V is called a disjunctive dominating set of G if for every vertex v ∈ V \D, v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by γ 2 (G). The MINIMUM DISJUNCTIVE DOMINATION PROBLEM (MDDP) is to find a disjunctive dominating set of cardinality γ 2 (G). Given a positive integer k and a graph G, the DISJUNCTIVE DOMINATION DECISION PROBLEM (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a (ln(∆ + ∆ + 2) + 1)-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within (1 − ) ln(|V |) for any > 0 unless NP ⊆ DTIME(|V | log |V |)). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.