Hop Domination in Graphs-II

Abstract

Let G = (V, E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γ h (G). In this paper we characterize the family of trees and unicyclic graphs for which γ h (G) = γt(G) and γ h (G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

Hop Dominating Sets in Graphs Under Binary Operations

“…Recently, Natarajan and Ayyaswamy [6] introduced and studied the concept of hop domination in a graph. In another study, Ayyaswamy et al [2] investigated the same concept and gave bounds of the hop domination number of some graphs.…”

“…In another study, Ayyaswamy et al [2] investigated the same concept and gave bounds of the hop domination number of some graphs. Henning and Rad [5] also studied the concept and answered a question posed by Ayyaswamy and Natarajan in [6]. They presented probabilistic upper bounds for the hop domination number and showed that the decision problems for the 2-step dominating set and hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.…”

A Note on the Hop Domination Number of a Subdivision Graph

“…Recently, Ayyaswamy and Natarajan ( [2,9]) initiated a study on a new domination parameter called hop domination number of a graph and characterized the family of trees and unicyclic graphs with equal hop domination number and total domination number. Ayyaswamy et al ( [1]) found some bounds on hop domination number of a tree.…”

On the complexity of some hop domination parameters

“…A subset S of vertices of a graph G is a hop dominating set (HDS) if every vertex outside S is at distance two from a vertex of S. The hop domination number, γ h (G), of G is the minimum cardinality of an HDS of G. An HDS of G of minimum cardinality is referred to a γ h (G)-set. The concept of hop domination was further studied, for example, in [1,4,7,9]. For a subset S ⊆ V (G) and a vertex v ∈ V (G), we say that v is hop dominated by S (or S hop dominates v) if either v ∈ S or v ∈ S and d(u, v) = 2 for some vertex u ∈ S. Ayyaswamy and Natarajan [9] also introduced the concept of hop independent domination in graphs.…”

“…The concept of hop domination was further studied, for example, in [1,4,7,9]. For a subset S ⊆ V (G) and a vertex v ∈ V (G), we say that v is hop dominated by S (or S hop dominates v) if either v ∈ S or v ∈ S and d(u, v) = 2 for some vertex u ∈ S. Ayyaswamy and Natarajan [9] also introduced the concept of hop independent domination in graphs. A subset S of vertices of a graph G is a hop independent dominating set (HIDS) if S is a HDS and for any pair v, w ∈ S, d(v, w) = 2.…”