Hop Differentiating Hop Dominating Sets in Graphs

Let G be a graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called an outer-convex hop dominating if for every two vertices x, y ∈ V (G) \ C, the vertex set of every x−y geodesic is contained in V (G) \ C and for every a ∈ V (G) \ C, there exists b ∈ C such that dG(a, b) = 2. The minimum cardinality of an outer-convex hop dominating set of G, denoted by ̃γconh(G), is called the outer-convex hop domination number of G. In this paper, we generate some formulas for the parameters of some special graphs and graphs under some binary operations by characterizing first the outer-convex hop dominating sets of each of thesegraphs. Moreover, we establish realization result that identifies and determines the connection of this parameter with the standard hop domination parameter. It shows that given any graph, this new parameter is always greater than or equal to the standard hop domination parameter.

Section: To See This Consider the Graphmentioning

confidence: 99%

Outer-Convex Hop Domination in Graphs Under Some Binary Operations

Isahac,

Hassan,

Laja

et al. 2023

“…The minimum cardinality among all hop dominating sets of G, denoted by γ h (G), is called the hop domination number of G. This parameter had studied on some families of graphs and graphs obtained from some operations in [1,2,9]. Researchers in the field had further investigated this concept, and introduced new variants and obtained some significant results that contributed a lot to the hop domination theory (see [3][4][5][6][7][8][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

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

Hassan,

Bakkang,

Sappari

2023

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 wedetermine 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.

“…Hedetniemi [9] in 2004. Thereafter, it has become an active research area (see [1,6,12,18,19,21,22,25,27]). It models many facility location problems (see [7]), where f (v) is viewed as cost function.…”

Section: Introductionmentioning

confidence: 99%

Hop Italian Domination in Graphs

Canoy Jr,

Jamil,

Menchavez

2023

Given a simple graph $G=(V(G),E(G))$, a function $f:V(G)\to \{0,1,2\}$ is a hop Italian dominating function if for every vertex $v$ with $f(v)=0$ there exists a vertex $u$ with $f(u)=2$ for which $u$ and $v$ are of distance $2$ from each other or there exist two vertices $w$ and $z$ for which $f(w)=1=f(z)$ and each of $w$ and $z$ is of distance $2$ from $v$. The minimum weight $\sum_{v\in V(G)}f(v)$ of a hop Italian dominating function is the hop Italian domination number of $G$, and is denoted by $\gamma_{hI}(G)$. In this paper, we initiate the study of the hop Italian domination. In particular, we establish some properties of the the hop Italian dominating function and explore the relationships of the hop Italian domination number with the hop Roman domination number \cite{Rad2,Natarajan} and with the $2$-hop domination number \cite{Canoy}. We study the concept under some binary graph operations. We establish tight bounds and determine exact values for their respective hop Italian domination numbers.