Global Hop Domination Numbers of Graphs

Abstract

A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. It is a global hop dominating set of G if it is a hop dominatingset of both G and the complement of G. The minimum cardinality of a global hop dominatingset of G, denoted by γgh(G), is called the global hop domination number of G. In this paper, we study the concept of global hop domination in graphs resulting from some binary operations.

Revisiting Domination, Hop Domination, and Global Hop Domination in Graphs

“…It is shown that the absolute difference of these parameters can be made arbitrarily large. Further, we rectify a result found in [16] and use the corrected result to prove some results in this paper.…”

“…Shortly after Natarajan and Ayyaswamy [13] introduced and studied the concept of hop domination in a graph, some variants of the concept emerged. The concept and some of its variants are studied in [5], [11], [12], [15], and [16]. In this paper, we show that the standard domination and hop domination parameters are generally non-comparable.…”

“…It was pointed out in [16] that 1 ≤ γ gh (G) ≤ |V (G)| for any graph G and that γ gh (G) = 1 if and only if G = K 1 . The next result is a rectification of Theorem 3.3 in [16].…”

On Resolving Hop Domination in Graphs

“…Natarajan and Ayyaswamy [9] introduced and studied the concept of hop domination in graphs. Hop domination in graphs were also studied in [6,10,11].…”

On Hop Domination Number of Some Generalized Graph Structures

“…Natarajan et al [13] found characterization results for hop domination number equals other domination parameters like total domination number, connected domination number for several families of graphs. Many scholars have explored this parameter in the years thereafter, leading to novel versions such as connected hop domination, total perfect hop domination, Roman hop domination, Global hop domination, etc., [11,12,[15][16][17][18]20]. In 2018, Natarajan et al [14] discussed hop domination number for some special families of graph like central graph, middle graph and total graph.…”