Algorithmic aspects of b-disjunctive domination in graphs

Abstract

For a fixed integer b>1 b>1 , a set D⊆V D⊆V is called a b-disjunctive dominating set of the graph G=(V,E) G=(V,E) if for every vertex v∈V∖D v∈V∖D , v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The Minimum b-Disjunctive Domination Problem (MbDDP) is to find a b-disjunctive dominating set of minimum cardinality. The cardinality of a minimum b-disjunctive dominating set of G is called the b-disjunctive domination number of G, and is denoted by γ d b (G) γbd(G) . Given a positive integer k and a graph G, the b-Disjunctive Domination Decision Problem (bDDDP) is to decide whether G has a b-disjunctive dominating set of cardinality at most k. In this paper, we first show that for a proper interval graph G, γ d b (G) γbd(G) is equal to γ(G) γ(G) , the domination number of G for b≥3 b≥3 and observe that γ d b (G) γbd(G) need not be equal to γ(G) γ(G) for b=2 b=2 . We then propose a polynomial time algorithm to compute a minimum cardinality b-disjunctive dominating set of a proper interval graph for b=2 b=2 . Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete even in chordal graphs. We also propose a (ln( Δ 2 +(b−1)Δ+b)+1) (ln⁡(Δ2+(b−1)Δ+b)+1) -approximation algorithm for MbDDP, where Δ Δ is the maximum degree of input graph G=(V,E) G=(V,E) and prove that MbDDP cannot be approximated within (1−ϵ)ln(|V|) (1−ϵ)ln⁡(|V|) for any ϵ>0 ϵ>0 unless NP ⊆ ⊆ DTIME (|V | O(loglog|V|) ) (|V|O(log⁡log⁡|V|)) . Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree max{b,4} max{b,4} .