Lexicographic Product Of Graphs Research Articles

Resolving Restrained Domination in Graphs

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

Universal lines in graphs

In a metric space M = (X, d), a line induced by two distinct points x, x′ ∈ X, denoted by , is the set of points given by A line is universal whenever . Chen and Chvátal [Disc. Appl. Math. 156 (2008), 2101-2108.] conjectured that in any finite metric space M = (X, d) either there is a universal line, or there are at least |X| different (nonuniversal) lines. A particular problem derived from this conjecture consists of investigating the properties of M that determine the existence of a universal line, and the problem remains interesting even if we can check that M has at least |X| different lines. Since the vertex set of any connected graph, equipped with the shortest path distance, is a metric space, the problem automatically becomes of interest in graph theory. In this paper, we address the problem of characterizing graphs that have universal lines. We consider several scenarios in which the study can be approached by analysing the existence of such lines in primary subgraphs. We first discuss the wide class of separable graphs, and then describe some particular cases, including those of block graphs, rooted product graphs and corona graphs. We also discuss important classes of nonseparable graphs, including Cartesian product graphs, join graphs and lexicographic product graphs.

From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of v∈V(G). Let f:V(G)→{0,1,2} be a function and Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1∪V2 has no isolated vertex and N(v)∩V2≠∅ for every v∈V(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=∑x∈V(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing γtRs(G) is NP-hard.

On the 2-Packing Differential of a Graph

Let G be a graph of order {text {n}}(G) and vertex set V(G). Given a set Ssubseteq V(G), we define the external neighbourhood of S as the set N_e(S) of all vertices in V(G){setminus } S having at least one neighbour in S. The differential of S is defined to be partial (S)=|N_e(S)|-|S|. In this paper, we introduce the study of the 2-packing differential of a graph, which we define as partial _{2p}(G)=max {partial (S):, Ssubseteq V(G) text { is a }2text {-packing}}. We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that partial _{2p}(G)+mu _{_R}(G)={text {n}}(G), where mu _{_R}(G) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on mu _{_R}(G), and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.

Star coloring under some graph operations

A star coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] such that no path of length [Formula: see text] in [Formula: see text] is bicolored. In this paper, the star chromatic number of join of graphs is computed. Some sharp bounds for the star chromatic number of corona, lexicographic, deleted lexicographic and hierarchical product of graphs together with a conjecture on the star chromatic number of lexicographic product of graphs are also presented.

From (Secure) w-Domination in Graphs to Protection of Lexicographic Product Graphs

Let $$w=(w_0,w_1, \dots ,w_l)$$ be a vector of nonnegative integers such that $$ w_0\ge 1$$ . Let G be a graph and N(v) the open neighbourhood of $$v\in V(G)$$ . We say that a function $$f: V(G)\longrightarrow \{0,1,\dots ,l\}$$ is a w-dominating function if $$f(N(v))=\sum _{u\in N(v)}f(u)\ge w_i$$ for every vertex v with $$f(v)=i$$ . The weight of f is defined to be $$\omega (f)=\sum _{v\in V(G)} f(v)$$ . Given a w-dominating function f and any pair of adjacent vertices $$v, u\in V(G)$$ with $$f(v)=0$$ and $$f(u)>0$$ , the function $$f_{u\rightarrow v}$$ is defined by $$f_{u\rightarrow v}(v)=1$$ , $$f_{u\rightarrow v}(u)=f(u)-1$$ and $$f_{u\rightarrow v}(x)=f(x)$$ for every $$x\in V(G){\setminus }\{u,v\}$$ . We say that a w-dominating function f is a secure w-dominating function if for every v with $$f(v)=0$$ , there exists $$u\in N(v)$$ such that $$f(u)>0$$ and $$f_{u\rightarrow v}$$ is a w-dominating function as well. The (secure) w-domination number of G, denoted by ( $$\gamma _{w}^s(G)$$ ) $$\gamma _{w}(G)$$ , is defined as the minimum weight among all (secure) w-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs $$G\circ H$$ are related to $$\gamma _w^s(G)$$ or $$\gamma _w(G)$$ . For the case of the secure domination number and the weak Roman domination number, the decision on whether w takes specific components will depend on the value of $$\gamma _{(1,0)}^s(H)$$ , while in the case of the total version of these parameters, the decision will depend on the value of $$\gamma _{(1,1)}^s(H)$$ .

Total colorings of certain classes of lexicographic product graphs

A total coloring of a graph G is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The Total Chromatic Number, [Formula: see text] is the minimum number of colors which need to be assigned to obtain a total coloring of the graph [Formula: see text]. The Total Coloring Conjecture made independently by Behzad and Vizing claims that, [Formula: see text], where [Formula: see text] represents the maximum degree of [Formula: see text]. The lower bound is sharp, the upper bound remains to be proved. In this paper, we prove the Total Coloring Conjecture for certain classes of lexicographic product and deleted lexicographic product of graphs.

Distance Laplacian spectra of joined union of graphs

The distance Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text] In this paper, we determine the distance Laplacian spectra of the graphs obtained by generalization of the join and lexicographic product of graphs (namely joined union). It is shown that the distance Laplacian spectra of these graphs not only depend on the distance Laplacian spectra of the participating graphs but also depend on the spectrum of another matrix of vertex-weighted Laplacian kind (analogous to the definition given by Chung and Langlands [A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996) 316–327]).

On independent position sets in graphs

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants.

Distance degree vector and scalar sequences of corona and lexicographic products of graphs with applications to dynamic NMR and dynamics of nonrigid molecules and proteins

We have developed novel topological techniques to generate distance degree vector sequences (DDS) and distance degree scalar sequences (SS) of corona and lexicographic products of graphs both of which are of chemical and biochemical interest. Query ID="Q1" Text="Please confirm if the author names are presented accurately and in the correct sequence (given name, middle name/initial, family name). Author 1 Given name: [Medha Itagi] Last name [Huilgol]. Also, kindly confirm the details in the metadata are correct." We have obtained exact analytical expressions for such products for a variety of graphs and trees of chemical and biochemical interest. We have also outlined the applications of these graph products and their DDS to dynamic NMR spectroscopy and isomerization/conformational graphs of nonrigid molecules and proteins whose NMR and conformational dynamics graphs are expressible as lexicographic products of smaller graphs.

A simple method for proving lower bounds in the zero-visibility cops and robber game

Cops and Robbers is a classical pursuit and evasion game in graph theory, which was introduced by Nowakowski and Winkler and independently by Quilliot. In this paper, we study the zero-visibility cops and robber game, which is a variant of Cops and Robbers. In the zero-visibility cops and robber game, the robber is invisible to the cops throughout the game. We introduce a simple method for proving lower bounds on the zero-visibility cop number. This lower bound method is based on a connection between the zero-visibility cop number and the matching number. Using this technique, we investigate graph joins, lexicographic products of graphs, complete multipartite graphs and split graphs. For each of these classes of graphs, we prove lower bounds and upper bounds on the zero-visibility cop number. We also present a linear time approximation algorithm for computing the lexicographic product of a tree and a graph G. The approximation ratio of this algorithm is bounded by $$|V(G)| / (\nu (G) + |V(G) {\setminus } V(\mathcal {M}(G))| )$$ , where V(G) is the vertex set of G, $$\nu (G)$$ is the matching number of G, $$\mathcal {M}(G)$$ is a maximum matching of G and $$V(\mathcal {M}(G))$$ is the vertex set of $$\mathcal {M}(G)$$ .

Clique doubly connected domination in the join and lexicographic product of graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

Antimagicness of Lexicographic Product Graph G[Pn

Hartsfield and Ringel conjectured that every connected graph other than K2 is antimagic. Since then, many classes of graphs have been proved to be antimagic. But few is known about the antimagicness of lexicographic product graphs. In this paper, via the construction of a directed Eulerian circuit, the Siamese method, and some modification on graph labeling, the antimagicness of lexicographic product graph G[Pn] is obtained.

Independent domination, colorings and the fractional idomatic number of a graph

Abstract A set S of vertices is an independent dominating set if it is both independent and dominating, and the idomatic number is the maximum number of vertex-disjoint independent dominating sets. In this paper we consider a fractional version of this. Namely, we define the fractional idomatic number as the maximum ratio | F | / m ( F ) over all families F of independent dominating sets, where m ( F ) denotes the maximum number of times an element appears in F . We start with some bounds including a connection with dynamic colorings. Then we show that the independent domination number of a planar graph with minimum degree 2 is at most half its order, and its fractional idomatic number is at least 2. Moreover, we show that an outerplanar graph of minimum degree 2 has idomatic number at least 2. We conclude by providing formulas for the parameters for the join, disjoint union and lexicographic product of graphs, while providing some bounds for cubic graphs.

English

English

Double domination in lexicographic product graphs

In a graph G, a vertex dominates itself and its neighbours. A subset S⊆V(G) is said to be a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality among all double dominating sets of G is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs G∘H in terms of invariants of the factor graphs G and H.

Total protection of lexicographic product graphs

Total protection of lexicographic product graphs

Some distance based indices of graphs based on four new operations related to the lexicographic product

For a (molecular) graph, the Wiener index, hyper-Wiener index and degree distance index are defined as $$W(G)= \sum_{\{u,v\}\subseteq V(G)}d_G(u,v),$$ $$WW(G)=W(G)+\sum_{\{u,v\}\subseteq V(G)} d_{G}(u,v)^2,$$ and $$DD(G)=\sum_{\{u,v\}\subseteq V(G)}d_G(u, v)(d(u/G)+d(v/G)),$$ respectively, where $d(u/G)$ denotes the degree of a vertex $u$ in $G$ and $d_G(u, v)$ is distance between two vertices $u$ and $v$ of a graph $G$. In this paper, we study Wiener index, hyper-Wiener index and degree distance index of graphs based on four new operations related to the lexicographic product, subdivision and total graph.

On Semitotal Domination in Graphs

A set $S$ of vertices of a connected graph $G$ is a semitotal dominating set if every vertex in $V(G)\setminus S$ is adjacent to a vertex in $S$, and every vertex in $S$ is of distance at most $2$ from another vertex in $S$. A semitotal dominating set $S$ in $G$ is a secure semitotal dominating set if for every $v\in V(G)\setminus S$, there is a vertex $x\in S$ such that $x$ is adjacent to $v$ and that $\left(S\setminus\{x\}\right)\cup \{v\}$ is a semitotal dominating set in $G$. In this paper, we characterize the semitotal dominating sets and the secure semitotal dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding semitotal domination and secure semitotal domination numbers.

Forcing Subsets for γc-sets and γt-sets in the Lexicographic Product of Graphs

In this paper, the connected dominating sets and total dominating sets in the lexicographic product of two graphs are characterized. Further, the connected domination, total domination, forcing connected domination and forcing total domination numbers of these graphs are determined.