Polynomial Time Algorithms for Hop Domination

Abstract

A set $S \subseteq V(G)$ is said to be a hop dominating set if every vertex $u \in V(G) \setminus S$, there exists a vertex $v \in S$ such that $d(u,v)=2$ where d(u, v) represents the distance between u and v in G. The minimum k for which there exists a hop dominating set of size k is called the hop domination number denoted by $\gamma _{h}(G)$. Henning et al. (Inf. Process. Lett. 2020) showed that Hop Domination  is NP-hard for bipartite graphs and chordal graphs. The following are the results of this paper.

• Henning et al. presented a linear time algorithm for solving Hop Domination on bipartite permutation graphs which is a proper subset of biconvex bipartite graphs. In this paper we present a polynomial algorithm for the problem on biconvex bipartite graphs, a superclass of bipartite permutation graphs.

• We show that Hop Domination is polynomial time solvable on interval graphs which are a subclass of chordal graphs.

• We initiate the study on this problem from the parameterized complexity perspective. We show that the decision version of Hop Domination is W[1]-hard when parameterized by solution size.