Lexicographic Product Of Graphs Research Articles

Omega Indices of Strong and Lexicographic Products of Graphs

Background: The degree sequence of a graph is the list of its vertex degrees arranged in usually increasing order. Many properties of the graphs realized from a degree sequence can be deduced by means of a recently introduced graph invariant called omega invariant. Methods: We used the definitions of the considered graph products together with the list of degree sequences of these graph products for some well-know graph classes. Naturally, the vertex degree and edge degree partitions are used. As the main theme of the paper is the omega invariant, we frequently used the definition and fundamental properties of this very new invariant for our calculations. Also, some algebraic properties of these products are deduced in line with some recent publications following the same fashion. Results: In this paper, we determine the degree sequences of strong and lexicographic products of two graphs and obtain the general form of the degree sequences of both products. We obtain a general formula for the omega invariant of strong and lexicographic products of two graphs. The algebraic structures of strong and lexicographic products are obtained. Moreover, we prove that strong and lexicographic products are not distributive over each other. Conclusion: We have obtained the general expression for degree sequences of two important products of graphs and a general expression for omega invariants of strong and lexicographic products. Furthermore, we have obtained algebraic structures of strong and lexicographic products in terms of their degree sequences. Also, it has been found that the disruptive property does not hold for strong and lexicographic products.

Computation of Weighted PI Index of Lexicographic product graphs and for Silicates Networks

The study of chemical compounds’ molecular structures is one of the most cutting-edge uses of graph theory, along with computer science, nanochemistry, network design in electrical and electronic engineering, and the depiction of graphs in Google Maps. The degree and distance between vertices of a graph are the basis for examining topological indices. The formula for computing the Weighted Padmakar Ivan index (WPI) of a graph G is PIw(G) = P e∈E(G) [(dG(u) + dG(v)][|V (G)| − NG(e)].

Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V ( G ) , let I ( S ) = ∪ u , v ∈ S I [ u , v ] . If I ( S ) = V ( G ) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g ( G ) . In this paper, the exact geodetic numbers of the product graphs T × K m and T ° K ¯ m are obtained, where T is a tree, K ¯ m denotes the complement of the complete graph K m and, × and ° denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

Differentiating Odd Dominating Sets in Graphs

Let $G=(V(G),E(G))$ be a simple and undirected graph. A dominating set \linebreak $S\subseteq V(G)$ is called a \textit{differentiating odd dominating set} if for every vertex $v\in V(G)$, \linebreak $|N[v]\cap S|\equiv 1(mod\ 2)$ and $N_G[u]\cap S\neq N_G[v]\cap S$ for every two distinct vertices $u$ and $v$ of $V(G)$. The minimum cardinality of a differentiating odd dominating set of $G$, denoted by $\gamma_D^o(G)$, is called the \textit{differentiating odd domination number}. In this paper, we discuss \linebreak differentiating odd dominating set in some graphs and give relationships between the \linebreak differentiating odd domination, odd domination, and differentiating-domination numbers. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of graphs and determine the differentiating odd domination numbers of these 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.

RETRACTED: On Vertex PI index of certain perfect graphs, lexicographic product of graphs, and some chemical networks

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433.

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

Identifying Secure Domination in the Cartesian and Lexicographic Products of Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in the Cartesian product and lexicographic product of two connected graphs.

The equidistant dimension of graphs: NP-completeness and the case of lexicographic product graphs

<abstract><p>Let $ V(G) $ be the vertex set of a simple and connected graph $ G $. A subset $ S\subseteq V(G) $ is a distance-equalizer set of $ G $ if, for every pair of vertices $ u, v\in V(G)\setminus S $, there exists a vertex in $ S $ that is equidistant to $ u $ and $ v $. The minimum cardinality among the distance-equalizer sets of $ G $ is the equidistant dimension of $ G $, denoted by $ \xi(G) $. In this paper, we studied the problem of finding $ \xi(G\circ H) $, where $ G\circ H $ denotes the lexicographic product of two graphs $ G $ and $ H $. The aim was to express $ \xi(G\circ H) $ in terms of parameters of $ G $ and $ H $. In particular, we considered the cases in which $ G $ has a domination number equal to one, as well as the cases where $ G $ is a path or a cycle, among others. Furthermore, we showed that $ \xi(G)\le \xi(G\circ H)\le \xi(G)|V(H)| $ for every connected graph $ G $ and every graph $ H $ and we discussed the extreme cases. We also showed that the general problem of finding the equidistant dimension of a graph is NP-hard.</p></abstract>

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.

Double total domination in the generalized lexicographic product of graphs

Let G be a graph of minimum degree at least two. A set D ⊆ V(G) is said to be a double total dominating set of G if |N (v) ∩ D| ≥ 2 for every vertex v ∈ V(G). The double total domination number of G, denoted by γ×2,t (G), is the minimum cardinality among all double total dominating sets of G. In this paper, we study this parameter for any generalized lexicographic product graph . In particular, we show that , where represents the total {2}-domination number of . Moreover, we obtain tight bounds and closed formulas on the double total domination number of this specific product of graphs in terms of domination invariants of the factor graphs. Finally, we characterize the graphs for which takes small values.

2-Locating Sets in a Graph

Let $G$ be an undirected graph with vertex-set $V(G)$ and edge-set $E(G)$, respectively. A set $S\subseteq V(G)$ is a $2$-locating set of $G$ if $\big|[\big(N_G(x)\backslash N_G(y)\big)\cap S] \cup [\big(N_G(y)\backslash N_G(x)\big)\cap S]\big|\geq 2$, for all \linebreak $x,y\in V(G)\backslash S$ with $x\neq y$, and for all $v\in S$ and $w\in V(G)\backslash S$, $\big(N_G(v)\backslash N_G(w)\big)\cap S \neq \varnothing$ or $\big(N_G(w)\backslash N_G[v]\big) \cap S\neq \varnothing$. In this paper, we investigate the concept and study 2-locating sets in graphs resulting from some binary operations. Specifically, we characterize the 2-locating sets in the join, corona, edge corona and lexicographic product of graphs, and determine bounds or exact values of the 2-locating number of each of these graphs.

Global Stable Location-Domination in Graphs

In this paper, we introduce and investigate the concept of global stable location-domination in graphs. We also characterize the global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs and determine the exact value or sharp bound of the corresponding global stable location-domination number.

$1$-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.

Split Restrained Geodetic Number of Strong Product and Lexicographic Product of Graphs

Split Restrained Geodetic Number of Strong Product and Lexicographic Product of Graphs

On Double Roman Dominating Functions in Graphs

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of 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.

From w-Domination in Graphs to Domination Parameters in Lexicographic Product Graphs

A wide range of parameters of domination in graphs can be defined and studied through a common approach that was recently introduced in [https://doi.org/10.26493/1855-3974.2318.fb9] under the name of w-domination, where w=(w_0,w_1, dots ,w_l) is a vector of non-negative integers such that w_0ge 1. Given a graph G, a function f: V(G)longrightarrow {0,1,dots ,l} is said to be a w-dominating function if sum _{uin N(v)}f(u)ge w_i for every vertex v with f(v)=i, where N(v) denotes the open neighbourhood of vin V(G). The weight of f is defined to be omega (f)=sum _{vin V(G)} f(v), while the w-domination number of G, denoted by gamma _{w}(G), is defined as the minimum weight among all w-dominating functions on G. A wide range of well-known domination parameters can be defined and studied through this approach. For instance, among others, the vector w=(1,0) corresponds to the case of standard domination, w=(2,1) corresponds to double domination, w=(2,0,0) corresponds to Italian domination, w=(2,0,1) corresponds to quasi-total Italian domination, w=(2,1,1) corresponds to total Italian domination, w=(2,2,2) corresponds to total {2}-domination, while w=(k,k-1,dots ,1,0) corresponds to {k}-domination. In this paper, we show that several domination parameters of lexicographic product graphs Gcirc H are equal to gamma _{w}(G) for some vector win {2}times {0,1,2}^{l} and lin {2,3}. The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H. In particular, we focus on quasi-total Italian domination, total Italian domination, 2-domination, double domination, total {2}-domination, and double total domination of lexicographic product graphs.

Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs.

Strong Resolving Domination in the Lexicographic Product of Graphs

Let G be a connected graph. A subset S ⊆ V (G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u, v ∈ V (G), there exists a vertex w ∈ S such that u ∈ IG[v, w] or IG[u, w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the lexicographic product of graphs and determine the corresponding resolving domination number.