Algorithm and hardness results on hop domination in graphs
Introduction
Two vertices u and v in a graph G are neighbors if they are adjacent, that is, . A set S of vertices of a graph G is a dominating set of G if every vertex in has a neighbor in S. The domination number, , is the minimum cardinality of a dominating set of G. The distance between two vertices u and v in a connected graph G, denoted , is the length of the shortest -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 , two vertices u and v in a graph G are said to k-step dominate each other if . A set in G is a k-step dominating set of G if every vertex in is k-step dominated by some vertex of S. The k-step domination number, , 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 of a graph G is a hop dominating set of G if every vertex of 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 .
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 .
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.
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In Section 3, we present a linear time algorithm for computing a minimum hop dominating set in bipartite permutation graphs.
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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.
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Finally, in Section 5, we present hardness results on Min-HDS. In particular, for a graph , we first show that Min-HDS cannot be approximated within a factor of for any , 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 . Finally, we show that Min-HDS is APX-complete for bipartite graphs of degree at most 3.
Section snippets
Terminology and notation
We use the standard notation . Let be a graph with vertex set and edge set . The open neighborhood of a vertex v in G is the set and the closed neighborhood of v is . The degree of a vertex v is and is denoted by . We simply use and if the context of the graph is clear. For a set S of vertices in G, the subgraph of G induced by S is denoted by . The distance between two vertices u and v, denoted by
Hop domination in bipartite permutation graphs
In this section, we present an time algorithm for computing a minimum hop dominating set in bipartite permutation graphs. If G is a disconnected graph having components where , then . Hence it suffices for us to consider only connected bipartite permutation graphs for the purpose of designing our algorithm.
Lemma 1 Suppose of is a forward-convex ordering of a connected bipartite permutation graph . If with
Hop domination in perfect elimination bipartite graphs
The decision version of Min-HDS is shown to be NP-complete for planar bipartite graphs [15] and hence for bipartite graphs. In this section, we show that the decision version of Min-HDS is NP-complete for perfect elimination bipartite graphs by providing a polynomial time reduction to it from the decision version of Min-HDS in bipartite graphs.
Theorem 3 The decision version of Min-HDS is NP-complete for perfect elimination bipartite graphs. Proof Clearly the decision version of Min-HDS is in NP. Given a
Hardness results
In this section, we discuss the lower bound and the upper bound of approximation ratio for Min-HDS in graphs. Unless otherwise stated, we adopt the terminology used in [2] throughout this section.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank the anonymous referees for their comments that lead to improvements in the paper.
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2021, Discussiones Mathematicae - Graph Theory
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Research supported in part by the University of Johannesburg.