Algorithm and hardness results on hop domination in graphs

Highlights

• A linear time algorithm is presented for computing a minimum hop dominating set in bipartite permutation graphs.

• Min-HDS is shown to be NP-hard for perfect elimination bipartite graphs.

• Min-HDS cannot be approximated within a factor of $(1-ϵ)\ln n$ for any $ϵ>0$, unless P=NP.

• Min-HDS is APX-complete for bipartite graphs of maximum degree 3.

Abstract

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph $G=(V,E)$ is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. Given a graph G, Min-HDS is the problem of finding a hop dominating set of G of minimum cardinality. The decision version of Min-HDS is known to be NP-complete for planar bipartite graphs and planar chordal graphs, and hence for bipartite graphs and chordal graphs. In this paper, we present a linear time algorithm for computing a minimum hop dominating set in bipartite permutation graphs, which is a subclass of bipartite graphs. We also show that Min-HDS cannot be approximated within a factor of $(1-\epsilon )\ln |V|$, unless P=NP and can be approximated within a factor of $1+\ln (\mathrm{\Delta }(\mathrm{\Delta }-1)+1)$, where Δ denotes the maximum degree in the graph G. Finally, we show that Min-HDS is APX-complete for bipartite graphs of maximum degree 3.

Introduction

Two vertices u and v in a graph G are neighbors if they are adjacent, that is, $uv\in E(G)$. A set S of vertices of a graph G is a dominating set of G if every vertex in $V(G)\setminus S$ has a neighbor in S. The domination number, $\gamma (G)$, is the minimum cardinality of a dominating set of G. The distance between two vertices u and v in a connected graph G, denoted $d_G(u,v)$, is the length of the shortest $u,v$-path in G. The fundamentals of domination theory and the concept of several domination parameters in graphs have been surveyed in the books [13], [14].

For an integer $k\ge 1$, two vertices u and v in a graph G are said to k-step dominate each other if $d_G(u,v)=k$. A set $S\subseteq V(G)$ in G is a k-step dominating set of G if every vertex in $V(G)$ is k-step dominated by some vertex of S. The k-step domination number, ${\gamma }_{\mathrm{kstep}}(G)$, of G, is the minimum cardinality of a k-step dominating set of G. The concept of 2-step domination in graphs was introduced and first studied by Chartrand, Harary, Hossain, and Schultz [7], and subsequently studied, for example, in [6], [11], [16].

Ayyaswamy and Natarajan [3] introduced a domination parameter of a graph closely related to the 2-step domination number, called hop domination in graphs. A set $S\subseteq V(G)$ of a graph G is a hop dominating set of G if every vertex of $V(G)\setminus S$ is 2-step dominated by some vertex of S. The minimum cardinality of a hop dominating set of a graph G is called the hop domination number of G and is denoted by ${\gamma }_{\mathrm{h}}(G)$.

Natarajan and Ayyaswamy [19] continued the further study on the equality of the hop domination number and other domination parameters. Ayyaswamy et al. [4] established upper and lower bounds on the hop domination number of a tree together with the characterization of extremal trees. Henning and Rad [15] present probabilistic upper bounds for the hop domination number of a graph. They also proved that the decision version of Min-HDS is NP-complete for planar bipartite graphs and planar chordal graphs. Chen and Wang [8] studied the relationship between the total domination number and the hop domination number in diamond-free graphs. Kundu and Majumder [17] gave a linear time algorithm to compute an optimal k-hop dominating set of a tree for $k\ge 1$.

In this paper, we continue the study on the algorithmic aspects of Min-HDS in graphs. The results presented in this paper are summarized as follows.

• In Section 3, we present a linear time algorithm for computing a minimum hop dominating set in bipartite permutation graphs.

• In Section 4, we show that the decision version of Min-HDS is NP-complete for perfect elimination bipartite graphs. We note that the class of perfect elimination bipartite graphs is a superclass of bipartite permutation graphs.

• Finally, in Section 5, we present hardness results on Min-HDS. In particular, for a graph $G=(V,E)$, we first show that Min-HDS cannot be approximated within a factor of $(1-\epsilon )\ln |V|$ for any $\epsilon >0$, unless P=NP. On the positive side, we show that for a graph G with maximum degree Δ, Min-HDS can be approximated within a factor of $1+\ln (\mathrm{\Delta }(\mathrm{\Delta }-1)+1)$. Finally, we show that Min-HDS is APX-complete for bipartite graphs of degree at most 3.