Dierentiating-Dominating sets in graphs Under binary operations

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

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ V (G) is a perfect hop dominating set of G if for every v ∈ V (G) \ S, there is exactly one vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of a perfect hop dominating set of G is called the perfect hop domination number of G, denoted by γph(G). A perfect hop dominating set S ⊆ V (G) is called a total perfect hop dominating set of G if for every v ∈ V (G), there is exactly one vertex u ∈ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by γtph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality γtph(G) is referred to as a γtph-set of G. In this paper, we characterize the total perfect hop dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding total perfect hop domination number.

A set $S\subseteq V(G)$ is a liar's dominating set ($lds$) of graph $G$ if $|N_G[v]\cap S|\geq 2$ for every $v\in V(G)$ and $|(N_G[u]\cup N_G[v])\cap S|\geq 3$ for any two distinct vertices $u,v \in V(G)$. The liar's domination number of $G$, denoted by $\gamma_{LR}(G)$, is the smallest cardinality of a liar's dominating set of $G$. In this paper we study the concept of liar's domination in the join, corona, and lexicographic product of graphs.

: There are four prominent product graphs in graph theory: Cartesian, strong, direct and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted G o H, for some digraphs G and H. The paper concludes by proving a cancellation property for the lexicographic product of digraphs G, H, A, and B: if G o H = A o B and /V(G)/ = /V(A)/, then G = A. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

. 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.

A lexicographic product of graphs G and H, denoted by is defined as a graph with the vertex set and an edge presents in the product whenever or (u 1 = u 2 and ). We investigate the sufficient conditions for vertex pancyclicity of lexicographic products of complete graphs Kn , paths Pn or cycles Cn with a general graph. We obtain that (i) if G 1 is a traceable graph of even order and G 2 is a graph with at least one edge, then is vertex pancyclic; (ii) if G 1 is hamiltonian and G 2 is a graph with at least one edge, then is vertex pancyclic.

A subset S of V (G), where G is a simple undirected graph, is hop dominating if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is hop differentiating if N2 G[u] ∩ S ̸= N2 G[v] ∩ S for any two distinct vertices u, v ∈ V (G). A set S ⊆ V (G) is hop differentiating hop dominating if it is both hop differentiating and hop dominating in G. The minimum cardinality of a hop differentiating hop dominating set in G, denoted by γdh(G), is called the hop differentiating hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the hop differentiating hop dominating sets in graphs under some binary operations.

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.

Given a graph \(G \), a set is \(\Delta \) convex if there is no vertex \(u\in V(G)\setminus S \) that forms a triangle with two vertices of \(S \). The \(\Delta \)-convex hull of \(S \) is the minimum \(\Delta \)-convex set containing \(S \). This article is an attempt to discuss the Carath\'eodory number and exchange number on various graph families and standard graph products namely Cartesian, strong and lexicographic products of graphs.

The oriented chromatic number of an oriented graph $\vec G$ is the minimum order of an oriented graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The oriented chromatic number of an undirected graph $G$ is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph $G$, defined as the minimum order of an oriented graph $\vev U$ such that every orientation $\vec G$ of $G$ admits a homomorphism to $\vec U$. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper oriented chromatic numbers of Cartesian, strong, direct and lexicographic products of graphs, and consider the particular case of products of paths.

Quasi-[formula omitted]-distance-balanced graphs

ABSTRACTLet be a symmetric matrix, or equivalently, a weighted graph whose edge ij has the weight . The eigenvalues of are the eigenvalues of M. We denote by the principal submatrix of M obtained by deleting from M both the ith row and the ith column. If μ is an eigenvalue of M, and thus of , of multiplicity , then vertex i of is a downer, or a neutral, or a Parter vertex, depending whether the multiplicity of μ in or, equivalently, in , is k−1, k, or k+1, respectively. In this paper, for a fixed μ, we consider vertex types according to the above classification in graphs which are generalized lexicographic products of an arbitrary graph over cliques and co-cliques, or connected regular graphs. In addition, we add some comments on constructions of large families of cospectral and integral 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.

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

The quadratic embedding constant (QE constant) of a graph is a new characteristic value of a graph defined through the distance matrix. We derive formulae for the QE constants of the join of two regular graphs, double graphs and certain lexicographic product graphs. Examples include complete bipartite graphs, wheel graphs, friendship graphs, completely split graph, and some graphs associated to strongly regular graphs.