$J^2$-Hop Domination in Graphs: Properties and Connections with Other Parameters

Abstract

A subset $T=\{v_1, v_2, \cdots, v_m\}$ of vertices of an undirected graph $G$ is called a $J^2$-set if
$N_G^2[v_i]\setminus N_G^2[v_j]\neq \varnothing$ for every $i\neq j$, where $i,j\in\{1, 2, \ldots, m\}$.
A $J^2$-set is called a $J^2$-hop dominating in $G$ if for every $a\in V(G)\s T$, there exists $b\in T$
such that $d_G(a,b)=2$. The $J^2$-hop domination number of $G$, denoted by $\gamma_{J^2h}(G)$, is the maximum
cardinality among all $J^2$-hop dominating sets in $G$. In this paper, we introduce this new parameter and we
determine its connections with other known parameters in graph theory. We derive its bounds with respect to
the order of a graph and other known parameters on a generalized graph, join and corona of two graphs. Moreover,
we obtain exact values of the parameter for some special graphs and shadow graphs using the characterization
results that are formulated in this study.