Algorithmic aspects of b-disjunctive domination in graphs

Abstract

For a fixed integer $b>1$, a set $D\subseteq V$ is called a b-disjunctive dominating set of the graph $G=(V,E)$ if for every vertex $v\in V{\setminus }D$, 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 $\gamma _{b}^{d}(G)$. 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, $\gamma _{b}^{d}(G)$ is equal to $\gamma (G)$, the domination number of G for $b \ge 3$ and observe that $\gamma _{b}^{d}(G)$ need not be equal to $\gamma (G)$ for $b=2$. We then propose a polynomial time algorithm to compute a minimum cardinality b-disjunctive dominating set of a proper interval graph for $b=2$. Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln ({\varDelta }^{2}+(b-1){\varDelta }+b)+1)$-approximation algorithm for MbDDP, where ${\varDelta }$ is the maximum degree of input graph $G=(V,E)$ and prove that MbDDP cannot be approximated within $(1-\epsilon ) \ln (|V|)$ for any $\epsilon >0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree $\max \{b,4\}$.