Lexicographic Product Research Articles

Unveiling new dimensions in soft disemigraph theory: lexicographic and restricted lexicographic products

Unveiling new dimensions in soft disemigraph theory: lexicographic and restricted lexicographic products

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.

Convex 2-Domination in Graphs

Let $G$ be a connected graph. A set $S \subseteq V(G)$ is \textit{convex $2$-dominating} if $S$ is both convex and $2$-dominating. The minimum cardinality among all convex $2$-dominating sets in $G$, denoted by $\gamma_{2con}(G)$, is called the \textit{convex $2$-domination number} of $G$. In this paper, we initiate the study of convex $2$- domination in graphs. We show that any two positive integers $a$ and $b$ with $6 \le a \le b$ are, respectively, realizable as the convex domination number and convex $2$-domination number of some connected graph. Furthermore, we characterize the convex $2$-dominating sets in the join, corona, lexicographic product, and Cartesian product of two graphs and determine the corresponding convex $2$-domination number of each of these graphs.

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.

On the skew eigenvalues of joined union of oriented graphs and applications

Let be an oriented graph with n vertices and m arcs having underlying graph G. The skew matrix of oriented graph , denoted by is a (−1, 0, 1)- skew symmetric matrix. The skew eigenvalues of are the eigenvalues of the matrix and its characteristic polynomial is the skew characteristic polynomial of . The sum of the absolute values of the skew eigenvalues is the skew energy of and is denoted by . In this paper, we extend the definition of joined union of graphs to oriented graphs. We show that the skew eigenvalues of the joined union of oriented graphs is the union of the skew eigenvalues of the component oriented graphs except some eigenvalues, which are given by an auxiliary matrix associated with the joined union. As a special case we obtain the skew eigenvalues of join of two oriented graphs and the lexicographic product of oriented graphs. We provide examples of orientations of some well known graphs to highlight the importance of our results. As applications to our result we obtain some new infinite families of skew equienergetic oriented graphs. Our results extend and generalize the results obtained in [H.S. Ramane, K.C. Nandeesh, I. Gutman and X. Li, Skew equienergetic digraphs, Trans. Comb., 5(1), (2016) 15-23].

Cardinal functions on lexicographic products

We will calculate the density, the spread and related cardinal functions on lexicographic products of GO-spaces, and give their applications.

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 de-gree 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 invari-ant, 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 prod-ucts in terms of their degree sequences. Also, it has been found that the disruptive property does not hold for strong and lexicographic products.

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

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

Secure Inverse Domination in the Corona and Lexicographic Product of Two Graphs

As secure domination and inverse domination garnered attention from various researchers, the combination of the two also raised a certain amount of curiosity. This paper aimed to investigate the secure inverse domination in graphs which is defined as follows. Let G be a connected simple graph and let D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is an inverse dominating set of G with respect to D. The set S is called a secure inverse dominating set of G if for every u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪{u} is a dominating set of G. The secure inverse domination number of G, denoted by γ_s^((-1) ) (G), is the minimum cardinality of a secure inverse dominating set of G. A secure inverse dominating set of cardinality γ_s^((-1) ) (G) is called γ_s^((-1) )-set. Particularly, the researchers examined and provided the characterization of secure inverse dominating set in the corona and lexicographic product of two graphs in this study. Moreover, the secure inverse domination number of graphs under the binary operations corona and lexicographic product were determined.

Outer-restrained Domination in the Lexicographic Product of Two Graphs

Let G be a connected simple graph. A set S⊆V(G) is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in (G)∖S . A set S of vertices of a graph G is an outer-restrained dominating set if every vertex not in S is adjacent to some vertex in S and V(G)∖S is a restrained set. The outer-restrained domination number of G, denoted by (γ_r ) ̃(G) is the minimum cardinality of an outer-restrained dominating set of G. An outer-restrained set of cardinality (γ_r ) ̃(G) will be called a (γ_r ) ̃(G) -set. This study is an extension of an existing research on outer-restrained domination in graphs. In this paper, we characterized the outer-restrained domination in graphs under the lexicographic product of two graphs.

Inverse Doubly Connected Domination in the Lexicographic Product of Two Graphs

Let G be a nontrivial connected graph. A dominating set D⊆V(G) is called a doubly connected dominating set of G if both 〈D〉 and 〈V(G)\D〉 are connected. Let D be a minimum connected dominating set of G. If S⊆V(G)\D is a connected dominating set of G, then S is called an inverse doubly connected dominating set of G with respect to D. Furthermore, the inverse doubly connected domination number, denoted by γ_cc^(-1) (G) is the minimum cardinality of an inverse doubly connected dominating set of G. An inverse doubly connected dominating set of cardinalities γ_cc^(-1) (G) is called γ_cc^(-1)-set. In this paper, we characterized the inverse doubly connected domination in the lexicographic product of two graphs and give some important results.

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.

Monitoring-edge-geodetic sets in product networks

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . For any e ∈ E ( G ) and u , v ∈ V ( G ) , the edge e is monitored by two vertices u and v in graph G if d G ( u , v ) ≠ d G − e ( u , v ) . A set M of vertices of G is a monitoring-edge-geodetic set of G if for any edge e ∈ E ( G ) there exists a pair u , v ∈ M such that e is monitored by u, v. The monitoring-edge-geodetic number meg ⁡ ( G ) is the cardinality of the minimum MEG-set in G. In this paper, we obtain the exact values or bounds for the MEG numbers of graph products, including join, corona, cluster, lexicographic products and direct products.

Aspekte van inligtingsonttrekkingstrukture in aanlyn woordeboeke

Aspects of information retrieval structures in online dictionaries. In printed dictionaries with a static article and book structure the relation between inserting data and retrieving information is not complex, and most dictionaries follow more or less the same ordering systems. Online dictionaries display dynamic structures with data often presented in various levels and compartments that compel the user to drill down (or vertically) into articles or laterally (or horizontally) within specific comments of articles, and to embark on search procedures that must negotiate new structures such as extended comments and even new search positions such as search tunnels. It is of paramount importance that users should be aware of the full spectrum of data included in a lexicographic product and should be able to select the data types of interest to them. Users of lexicographic tools need assistance to ensure an optimal retrieval of information from the data. Consequently, lexicographers need to embark on dedicated ways to establish an information retrieval structure that can enable users to achieve an adequate dictionary consultation. This might demand a refinement of the database of the dictionary as well as a sophisticated interface that can provide new features and navigation options to the user. It emphasizes yet again the need for a multidisciplinary team in the development of a dictionary to provide the user with an appropriate final product. This paper offers proposals for structured and rapid access to data and the use of data indicators to guide users to items occurring in complex dictionary articles. The emphasis is also on user-friendliness of lexicographic instruments that can enhance the quality of intuitive dictionary use. Keywords: article structure, compartments, comprehensive article, contextualisation, information retrieval structure, levels, linking, online dictionaries, restricted article, search tunnel, search zone, subcomment on semantics

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>

Edge-coloring of generalized lexicographic product of graphs

<abstract><p>An edge-coloring of a graph $ G $ is an assignment of colors to its edges so that no two edges incident to the same vertex receive the same color. The chromatic index of $ G $, denoted by $ \chi'(G) $, is the least $ k $ for which $ G $ has a $ k $ edge-coloring. Graphs with $ \chi'(G) = \Delta(G) $ are said to be Class 1, and graphs with $ \chi'(G) = \Delta(G)+1 $ are said to be Class 2. Let $ G $ be a graph with $ V(G) = \{t_1, t_2, \ldots, t_n\} $, $ n\geq2 $, and $ h_n = (H_i)_{i\in\{1, 2, \ldots, n\}} $ be a sequence of vertex-disjoint graphs with $ V(H_i) = \{(t_i, y_1), (t_i, y_2), \ldots, (t_i, y_{m_i})\} $, $ m_i\geq1 $. The generalized lexicographic product $ G[h_n] $ of $ G $ and $ h_n = (H_i)_{i\in\{1, 2, \ldots, n\}} $ is a simple graph with vertex set $ \bigcup_{i = 1}^{n}V(H_i) $, in which $ (t_i, y_p) $ is adjacent to $ (t_j, y_q) $ if and only if either $ t_i = t_j $ and $ (t_i, y_p)(t_i, y_q)\in E(H_i) $ or $ t_it_j\in E(G) $. If $ G $ is a complete graph with order 2, then $ G[h_2] $ denotes a join $ H_1+H_2 $ of vertex-disjoint graphs $ H_1 $ and $ H_2 $. If $ H_i\cong H $ for $ i = 1, 2, \ldots, n $, then $ G[h_n] = G[H] $, where $ G[H] $ denotes the lexicographic product of two graphs $ G $ and $ H $. In this paper, we provide sufficient conditions for the generalized lexicographic product $ G[h_n] $ of $ G $ and $ h_n = (H_i)_{i\in\{1, 2, \ldots, n\}} $ to be Class 1, where all graphs in $ h_n $ have the same number of vertices.</p></abstract>

Interval edge colorings of the generalized lexicographic product of some graphs

<p>An edge-coloring of a graph $ G $ with colors $ 1, \ldots, t $ is an interval $ t $-coloring if all colors are used and the colors of edges incident to each vertex of $ G $ are distinct and form an interval of integers. A graph $ G $ is interval colorable if it has an interval $ t $-coloring for some positive integer $ t $. For an interval colorable graph $ G $, the least and the greatest values of $ t $ for which $ G $ has an interval $ t $-coloring are denoted by $ w(G) $ and $ W(G) $. Let $ G $ be a graph with vertex set $ V(G) = \{u_1, \ldots, u_{m}\} $, $ m\geq2 $, and let $ h_m = (H_i)_{i\in\{1, \ldots, m\}} $ be a sequence of vertex-disjoint with $ V(H_i) = \{x_1^{(i)}, \ldots, x_{n_i}^{(i)}\} $, $ n_i\geq 1 $. The generalized lexicographic products $ G[h_m] $ of $ G $ and $ h_m $ is a simple graph with vertex set $ \cup_{i = 1}^{m}V(H_i) $, in which $ x_p^{(i)} $ is adjacent to $ x_q^{(j)} $ if and only if either $ u_i = u_j $ and $ x_p^{(i)}x_q^{(i)}\in E(H_i) $ or $ u_iu_j\in E(G) $. In this paper, we obtain several sufficient conditions for the generalized lexicographic product $ G[h_m] $ to have interval colorings. We also present some sharp bounds on $ w(G[h_m]) $ and $ W(G[h_m]) $ of $ G[h_m] $.</p>

The general position avoidance game and hardness of general position games

Given a graph G, a set S of vertices in G is a general position set if no triple of vertices from S lie on a common shortest path in G. The general position achievement/avoidance game is played on a graph G by players A and B who alternately select vertices of G. A selection of a vertex by a player is a legal move if it has not been selected before and the set of selected vertices so far forms a general position set of G. The player who picks the last vertex is the winner in the general position achievement game and is the loser in the avoidance game. In this paper, we prove that the general position achievement/avoidance games are PSPACE-complete even on graphs with diameter at most 4. For this, we prove that the misère play of the classical Node Kayles game is also PSPACE-complete. As positive results, we obtain linear time algorithms to decide the winning player of the general position avoidance game in rook's graphs, grids, cylinders, and lexicographic products with complete second factors.