Hop Domination Research Articles

Connected Co-Independent Hop Domination in the Edge Corona and Complementary Prism of Graphs

Let $G$ be a connected graph. A subset $S$ of $V(G)$ is a connected co-independent hop dominating set in $G$ if the subgraph induced by $S$ is connected and $V(G) \backslash S$ is an independent set where for each $v \in V(G) \backslash S$, there exists a vertex $u \in S$ such that $d_G(u,v)=2$. The smallest cardinality of such an $S$ is called the connected co-independent hop domination number of $G$. This paper presents the characterizations of the connected co-independent hop dominating sets in the edge corona and complementary prism of graphs and determines the exact values of their corresponding connected co-independent hop domination number.

Closed Geodetic Hop Domination in Graphs

Let G be a simple, undirected and connected graph. A subset S ⊆ V (G) is a geodetic cover of G if IG[S] = V (G), where IG[S] is the set of all vertices of G lying on any geodesic between two vertices in S. A geodetic cover S of G is a closed geodetic cover if the vertices in S are sequentially selected as follows: Select a vertex v1 and let S1 = {v1}. If G is nontrivial, select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Where possible, for i ≥ 3, successively select vertex vi ∈/ IG[Si−1] and let Si = {v1, v2, ..., vi}. Then there exists a positive integer k such that Sk = S. A geodetic cover S of G is a geodetic hop dominating set if every vertex in V (G) \ S is of distance2 from a vertex in S. A geodetic hop dominating set S is a closed geodetic hop dominating set if S is a closed geodetic cover of G. The minimum cardinality of a (closed) geodetic hop dominating set of G is the (closed) geodetic hop domination number of G. This study initiates the study of the closed geodetic hop domination. First, it characterizes all graphs G of order n whose closed geodetic hop domination numbers are 2 or n, and determines the closed geodetic hop domination number of paths, cycles and multigraphs. Next, it shows that any positive integers a and b with 2 ≤ a ≤ b are realizable as the closed geodetic number and closed geodetic hop domination number of a connected graph. Also, every positive integer n, m and k with 4 ≤ m ≤ k and 2k−m+2 ≤ n are realizable as the order, geodetic hop domination number and closed geodetic hop domination number, respectively of a connected graph. Furthermore, the study characterizes the closed geodetic hop dominating sets of graphs resulting from the join, corona and edge corona of graphs.

On Minimal Geodetic Hop Domination in Graphs

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set $S \subseteq V(G)$ is a geodetic hop dominating set of $G$ if the following two conditions hold for each $x\in V(G)\setminus S$: $(1)$ $x$ lies in some $u$-$v$ geodesic in $G$ with $u,v\in S$, and $(2)$ $x$ is of distance $2$ from a vertex in $S$. The minimum cardinality $\gamma_{hg}(G)$ of a geodetic hop dominating set of $G$ is the geodetic hop domination number of $G$. A geodetic hop dominating set $S$ is a minimal geodetic hop dominating set if $S$ does not contain a proper subset that is itself geodetic hop dominating set. The maximum cardinality of a minimal geodetic hop dominating set in $G$ is the \emph{upper geodetic hop domination number} of $G$, and is denoted by $\gamma^{+}_{hg}(G)$. This paper initiates the study of the minimal geodetic hop dominating set and the corresponding upper geodetic hop domination number of nontrivial connected graphs. Interestingly, every pair of positive integers $a$ and $b$ with $2\le a\le b$ is realizable as the geodetic domination number and the upper geodetic hop domination number, respectively, of some graph. Furthermore, this paper investigates the concept in the join, corona and lexicographic product of graphs.

Further results on the hop domination number of a graph

A hop dominating set S in a connected graph G is called a minimal hop dominating set if no proper subset of S is a hop dominating set of G. The upper hop domination number γ+h (G) of G is the maximum cardinality of a minimal hop dominating set of G. Some general properties satisfied by this concept are studied. It is shown that for every two positive integers a and b where 2 ≤ a ≤ b, there exists a connected graph G such that γh(G) = a and γ+ h (G) = b. It is proved that minimal hop dominating set is NP-complete. It is proved that γh(G) and γ(G) are in general incomparable. It is shown that for every pair of positive integers a and b with a ≥ 2 and b ≥ 1, there exists a connected graph G such that γh(G) = a and γ(G) = b. We present an algorithm to compute minimal hop dominating set of G. Finally, we formulate an Integer linear programming problem to compute the hop domination number of G.

2-distance Zero Forcing Sets in Graphs

In this paper, we introduce new concept in graph theory called 2-distance zero forcing. We give some properties of this new parameter and investigate its connections with other parameters such as zero forcing and hop domination. We show that 2-distance zero forcing and hop domination (respectively, zero forcing parameter) are incomparable. Moreover, we characterize 2-distance zero forcing sets in some special graphs, and finally derive the exact values or bounds of the parameter using these results.

Vertex Cover Hop Dominating Sets in Graphs

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Some Properties of Zero Forcing Hop Dominating Sets in a Graph

In this paper, we initiate the study of a zero forcing hop domination in a graph. We establish some properties of this parameter and we determine its connections with other known parameters in graph theory. Moreover, we obtain some exact values or bounds of the parameter on the generalized graph, some families of graphs, and graphs under some operations via characterizations.

Transversal total hop domination in graphs

Let [Formula: see text] be a finite simple graph. A set [Formula: see text] is a total hop dominating set of [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. Any total hop dominating set of [Formula: see text] of minimum cardinality is a [Formula: see text]-set of [Formula: see text]. A total hop dominating set [Formula: see text] of [Formula: see text] which intersects every [Formula: see text]-set of [Formula: see text] is a transversal total hop dominating set. The minimum cardinality [Formula: see text] of a transversal total hop dominating set in [Formula: see text] is the transversal total hop domination number of [Formula: see text]. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph [Formula: see text] of order [Formula: see text] for which [Formula: see text] is [Formula: see text] or [Formula: see text], and also we determine the specific values of [Formula: see text] for some special graphs [Formula: see text]. Next, we solve some realization problems involving [Formula: see text] with other parameters of [Formula: see text]. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.

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

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.

Outer-Convex Hop Domination in Graphs Under Some Binary Operations

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.

Grundy Total Hop Dominating Sequences in Graphs

Let G = (V (G), E(G)) be an undirected graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Sˆ ={v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2G(vi) \Si−1j=1 N2G(vj ) ̸= ∅for every i ∈ {2, . . . , k}. If, in addition, Sˆ is a total hop dominating set of G, then S is a Grundy total hop dominating sequence. The maximum length of a Grundy total hop dominating sequence in a graph G, denoted by γth gr(G), is the Grundy total hop domination number of G. In this paper, we show that the Grundy total hop domination number of a graph G is between the total hop domination number and twice the Grundy hop domination number of G. Moreover, determine values or bounds of the Grundy total hop domination number of some graphs.

Hop Italian Domination in Graphs

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.

Another Look at Geodetic Hop Domination in a Graph

Let $G$ be an undirected graph with vertex and edge sets $V(G)$ and $E(G)$, respectively. A subset $S$ of vertices of $G$ is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number of $G$ is the minimum cardinality among all geodetic hop dominating sets in $G$. Geodetic hop dominating sets in a graph resulting from the join of two graphs have been characterized. These characterizations have been used to determine the geodetic hop domination number of the graphs considered. A realization result involving the hop domination number and geodetic hop domination number is also obtained.

Some Properties and Realization Problems Involving Connected Outer-hop Independent Hop Domination in Graphs

In this paper, we construct a realization problems involving connected outer-hop independent hop domination and we determine its connections with other known parameters in graph theory. In particular, given two positive integers a and b with 2 ≤ a ≤ b are realizable as the connected hop domination, connected outer-hop independent hop domination, and connected outer-independent hop domination numbers, respectively, of a connected graph. In addition, we characterize the connected outer-hop independent hop dominating sets in some families of graphs, join and corona of two graphs, and we use these results to derive formulas for the parameters of these graphs.

$1$-movable $2$-Resolving Hop Domination in Graph

Let $G$ be a connected graph. A set $S$ of vertices in $G$ is a 1-movable 2-resolving hop dominating set of $G$ if $S$ is a 2-resolving hop dominating set in $G$ and for every $v \in S$, either $S\backslash \{v\}$ is a 2-resolving hop dominating set of $G$ or there exists a vertex $u \in \big((V (G) \backslash S) \cap N_G(v)\big)$ such that $\big(S \backslash \{v\}\big) \cup \{u\}$ is a 2-resolving hop dominating set of $G$. The 1-movable 2-resolving hop domination number of $G$, denoted by $\gamma^{1}_{m2Rh}(G)$ is the smallest cardinality of a 1-movable 2-resolving hop dominating set of $G$. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the 1-movable 2-resolving hop dominating sets in the join, corona and lexicographic products of graphs, and determine the bounds of the 1-movable 2-resolving hop domination number of each of these graphs.

Connected Grundy Hop Dominating Sequences in Graphs

In this paper, we introduce and do an initial investigation of a variant of Grundy hop domination in a graph called the connected Grundy hop domination. We show that the connected Grundy hop domination number lies between the connected hop domination and Grundy hop domination number of a graph. In particular, we give realization results involving connected hop domination, connected Grundy hop domination, and Grundy hop domination numbers. Moreover, we determine the connected Grundy hop domination numbers of some graphs.

Outer-Connected 2-Resolving Hop Domination in Graphs

. Let G be a connected graph. A set S ⊆ V (G) is an outer-connected 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-resolving hop domination number of G, denoted by γ^c2Rh(G) is the smallest cardinality of an outer-connected 2-resolving hop dominating set of G. This study aims to combine the concept of outer-connected hop domination with the 2-resolving hop dominating sets of graphs. The main results generated in this study include the characterization of outer-connected 2-resolving hop dominating sets in the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values.

Grundy Dominating and Grundy Hop Dominating Sequences in Graphs: Relationships and Some Structural Properties

In this paper, we revisit the concepts of Grundy domination and Grundy hop domination in graphs and give some realization results involving these parameters. We show that the Grundy domination number and Grundy hop domination number of a graph G are generally incomparable (one is not always less than or equal the other). It is shown that their absolute difference can be made arbitrarily large. Moreover, the Grundy domination numbers of some graphs are determined.

On 1-movable Strong Resolving Hop Domination in Graphs

A set S is a 1-movable strong resolving hop dominating set of G if for every v ∈ S, either S\{v} is a strong resolving hop dominating set or there exists a vertex u ∈ (V (G)\S)∩NG(v) such that (S \ {v}) ∩ {u} is a strong resolving hop dominating set of G. The minimum cardinality of a 1-movable strong resolving hop dominating set of G is denoted by γ 1 msRh(G). In this paper, we obtained the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolving hop dominating sets in these types of graphs and determine the bounds or exact values of their 1-movable strong resolving hop domination numbers.

Weakly Convex Hop Dominating Sets in Graphs

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called weakly convex hop dominating if for every two vertices x, y ∈ C, there exists an x-y geodesic P(x, y) such that V (P(x, y)) ⊆ C and for every v ∈ V (G)\C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of a weakly convex hop dominating set of G, denoted by γwconh(G), is called the weakly convex hop domination number of G. In this paper, we introduce and initially investigate the concept of weakly convex hop domination. We show that every two positive integers a and b with 3 ≤ a ≤ b are realizable as the weakly convex hop domination number and convex hop domination number of some connected graph. Furthermore, we characterize the weakly convex hop dominating sets in some graphs under some binary operations.