52,109 Results Found

Determination of the relationships between environmental factors and diatom assemblages is usually made for several hundred lakes spread over a large area. However, the analysis of several lakes located near Lake Wigry also gives interesting results....

In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differentia...

In this article, we present a novel framework for edge domination in vague incidence graphs (VIG). We introduce the notion of certain types of vague incidence graphs and extend the concepts of dominations into edge and incidence domination in VIG. In...

A vertex subset D of G is a dominating set if every vertex in $V\left(G\right)\backslash D$ is adjacent to a vertex in D. A dominating set D is independent if $G\left[D\right]$, the subgraph of G induced by D, contains no edge. The domination number $\gamma \left(G\right)$ of a graph G is the minim...

Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized g...

The notion of Roman $\left\{2\right\}$-domination was introduced in 2016 as a variant of Roman domination, a concept inspired by a defending strategy used by the emperor Constantine (272–337 AD) to protect the Roman Empire. Since then, a considerable number o...

A dominating set is a classic concept that is widely used in road safety, disaster rescue operations, and chemical graphs. In this paper, we introduce a variation of the dominating set: the proper 3-dominating set. For a proper 3-dominating set D of...

This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graph...

Let $\mathcal{D}=\left(V\right(\mathcal{D}),A(\mathcal{D}\left)\right)$ be a finite, simple digraph and k a positive integer. A function $f:V\left(\mathcal{D}\right)\to \{0,1,2,\dots ,k+1\}$ is called a $\left[k\right]$-Roman dominating function (for short, $\left[k\right]$-RDF) if $f\left(A{N}^{-}\left[v\right]\right)\ge \left|A{N}^{-}\left(v\right)\right|+k$ for any vertex $v\in V\left(\mathcal{D}\right)$, wher...

African swine fever (ASF) caused by ASF virus (ASFV) is a fatal disease in pigs and results in great economic losses. Due to the lack of available vaccines and treatments, serological diagnosis of ASF plays a key role in the surveillance program, but...

This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where $w=(w_0,w_1,\&he$...

A vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A set S of vertices in a graph G is a total dominating set of G if every vertex of G is totally dominated by at least one vertex of S. For a total dominating set S of a graph...

Since 2016, when the first paper of the double Roman domination appeared, the topic has received considerable attention in the literature. We survey known results on double Roman domination and some variations of the double Roman domination, and a li...

A dominating set in a graph G is a set of vertices $S\subseteq V(G)$ such that any vertex of $V-S$ is adjacent to at least one vertex of $S.$ A dominating set S of G is said to be a perfect dominating set if each vertex in...

For a graph $G=(V,E)$ with vertex set $V=V\left(G\right)$ and edge set $E=E\left(G\right)$ , a Roman $\left\{3\right\}$ -dominating function (R $\left\{3\right\}$ -DF) is a function $f:V\left(G\right)\to \{0,1,2,3\}$ having the property that...

A vertex–edge Roman $\left\{2\right\}$-dominating function on a graph $G=(V,E)$ is a function $f:V⟶\{0,1,2\}$ satisfying that, for every edge $uv\in E$ with $f\left(v\right)=f\left(u\right)=0$, ${\sum }_{w\in N\left(v\right)\cup N\left(u\right)}f\left(w\right)\ge 2$. The weight of the function f is the sum ${\sum }_{a\in }$...

Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuz...

Let G be a graph with no isolated vertex and let $N\left(v\right)$ be the open neighbourhood of $v\in V\left(G\right)$. Let $f:V\left(G\right)\to \{0,1,2\}$ be a function and $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. We say that f is a strongly total Roman dominating function on G if the subgraph i...

Traditional stochastic dominance rules are so strict and qualitative conditions that generally a stochastic dominance relation between two alternatives does not exist. To solve the problem, we firstly supplement the definitions of almost stochastic d...

Given a graph $G=(V,E)$ , a function $f:V\to \{0,1,2,\cdots \}$ is said to be a total dominating function if ${\sum }_{u\in N\left(v\right)}f\left(u\right)>0$ for every $v\in V$ , where $N\left(v\right)$ denotes the op...

A total dominating set D of a graph G is said to be a secure total dominating set if for every vertex $u\in V\left(G\right)\backslash D$ , there exists a vertex $v\in D$ , which is adjacent to u, such that $(D\backslash \{v\left\}\right)\cup \left\{u\right\}$ is a tot...

A domination coloring of a graph G is a proper vertex coloring of G, such that each vertex of G dominates at least one color class (possibly its own class), and each color class is dominated by at least one vertex. The minimum number of colors among...

For a connected graph $G=(V,E)$, the dominating set in graph G is a subset of vertices $F\subset V$ such that every vertex of $V-F$ is adjacent to at least one vertex of F. The minimum cardinality of a dominating set of G, denoted by $\gamma \left(G\right)$, is the...

The domination problem and three of its variants (total domination, 2-domination, and secure domination) are considered. These problems have various real-world applications, including error correction codes, ad hoc routing for wireless networks, and...

Let G be a graph and $S\subseteq V\left(G\right).$ If S is a dominating set of G, and for each vertex $u\in V\left(G\right)-S,$ there is a neighbor of u in $S,$ denoted by $v,$ such that $(S\setminus \{v\left\}\right)\cup \left\{u\right\}$ is a dominating set of $G,$ then S is a secure dominating set (SDS) o...

A total perfect Roman dominating function (TPRDF) on a graph $G=(V,E)$ is a function f from V to $\{0,1,2\}$ satisfying (i) every vertex v with $f\left(v\right)=0$ is a neighbor of exactly one vertex u with $f\left(u\right)=2$; in addition, (ii) the subgraph of G that is induced by...

Perfect Roman Dominating Functions and Unique Response Roman Dominating Functions are two ways to translate perfect code into the framework of Roman Dominating Functions. We also consider the enumeration of minimal Perfect Roman Dominating Functions...

Handgrip strength (HGS) is a key indicator of muscular fitness and physical health. Recently, attention has focused on HGS asymmetry as a potential marker of functional impairment. The aim of the study was to analyze differences in handgrip strength...

A total k-rainbow dominating function on a graph $G=(V,E)$ is a function $f:V\left(G\right)\to 2^{\{1,2,\dots ,k\}}$ such that (i) ${\cup }_{u\in N\left(v\right)}f\left(u\right)=\{1,2,\dots ,k\}$ for every vertex v with $f\left(v\right)=\varnothing$, (ii) ${\cup }_{u\in N\left(v\right)}f\left(u\right)\ne \varnothing$ for $f\left(v\right)\ne \varnothing$. T...

Let G be a graph without isolated vertices. A function $f:V\left(G\right)\to \{0,1,2\}$ is a total Roman dominating function on G if every vertex $v\in V\left(G\right)$ for which $f\left(v\right)=0$ is adjacent to at least one vertex $u$...

Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In su...

For a vertex subset S of a graph $G,$ if each vertex of G is either in S or adjacent to some vertex in $S,$ then S is a dominating set of $G.$ Let S be a dominating set of a graph $G.$ If each vertex v not in S has a neighbor u in S such that $(S\backslash \{u\left\}\right)\cup \left\{v\right\}$...

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue “Advances in graph theory and Symmetry/Asymmetry” of Symmetry. Symmetry is one of the most important criteria that...

Let $G=(V,E)$ be a graph; a set $D\subseteq V$ is a total dominating set if every vertex $v\in V$ has, at least, one neighbor in D. The total domination number ${\gamma }_t\left(G\right)$ is the minimum cardinality among all total dominating sets. Given an arbitrary graph G...

A vertex v of a graph $G=(V,E)$ , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is a double vertex-edge dominating set if every edge of E is ve-dominated by at least...

Let R be a finite commutative ring with nonzero identity 1. In this paper, domination parameters, the domination number and the total dominating number, of the unit graph $\mathrm{G}(R)$ or the closed unit graph $\overline{\mathrm{G}}(R)$ of R are investigated. To study the do...

Eye dominance is often defined as a preference for the visual input of one eye to the other. Implicit in this definition is the dominant eye has better visual function. Several studies have investigated the effect of visual direction or defocus on oc...

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γ_pr(G) of G. Th...

Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, co...

In this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from $\{0,1,2,3,4\}$ to the vertices of a graph such that every vertex is protected by at least three units either on itself o...

Stroke is one of the leading causes of human disability globally. Motor function deficits resulting from a stroke affect the entire body, but relatively often it is the upper limbs that remain ineffective, which is very limiting in everyday life acti...

A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number ${\gamma }_{kt}\left(G\right)$ is the cardinality...

This paper establishes fundamental results for partial total domination in hypergraphs. We present tight bounds for the partial total domination number in k-uniform hypergraphs, demonstrate relationships with classical domination parameters, and prov...

During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there ar...

The notion of fuzzy graph (FG) is widely used in many problems arising from partial or incomplete descriptions of the real world and in particular from fields such as engineering, economics, computer science, social disciplines, or medical diagnostic...

Let G = (V, E) be a graph, where D is a subset of V, such that the set of vertices in the set (V-D) is adjacent to at least one vertex in set D. Then, set D is known as the domination set. In other words, a dominating set in graph G is a set of verti...

Let G be a graph with no isolated vertex and $f:V\left(G\right)\to \{0,1,2\}$ a function. Let $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. The function f is an outer-independent Roman dominating function on G if $V_0$ is an independent set and every vertex in $V_{}$...

Graphs are one of the dynamic tools used to solve network-related problems and real-time application models. The stability of the network plays a crucial role in ensuring uninterrupted data flow. A network becomes vulnerable when a node or a link bec...

We have studied the k-rainbow domination number of $C_n\square C_m$ for $k\ge 4$ (Gao et al. 2019), in which we present the 3-rainbow domination number of $C_n\square C_m$ , which should be bounded above by the four-rainbow dominatio...

In a digraph $D=\left(V\right(D),A(D\left)\right)$ , a two-valued function $f:V\left(D\right)\to \{-1,1\}$ defined on the vertices of D is called a signed dominating function if $f\left(N^{-}\left[v\right]\right)\ge 1$ for every v in D. The w...