238 research outputs found
From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs
[EN] Let G be a graph with no isolated vertex and let N (v) be the open neighbourhood of v is an element of V (G). Let f : V (G) -> {0, 1, 2} be a function and V-i = {v is an element of V (G) : f (v) = i} for every i is an element of{0, 1, 2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V-1 boolean OR V-2 has no isolated vertex and N (v) boolean AND V-2 not equal empty set for every v is an element of V (G) \ V2. The strongly total Roman domination number of G, denoted by gamma(s)(tR) (G), is defined as the minimum weight omega(f) = Sigma(x is an element of 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 gamma(s)(tR) (G) is NP-hard.Almerich-Chulia, A.; Cabrera Martinez, A.; Hernandez Mira, FA.; Martín Concepcion, PE. (2021). From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs. Symmetry (Basel). 13(7):1-10. https://doi.org/10.3390/sym13071282S11013
On the Roman domination in the lexicographic product of graphs
AbstractA Roman dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex with f(v)=0 is adjacent to some vertex with f(v)=2. The Roman domination number of G is the minimum of w(f)=∑v∈Vf(v) over all such functions. Using a new concept of the so-called dominating couple we establish the Roman domination number of the lexicographic product of graphs. We also characterize Roman graphs among the lexicographic product of graphs
Total protection in graphs
Suposem que una o diverses entitats estan situades en alguns dels vèrtexs d'un graf simple, i que una entitat situada
en un vèrtex es pot ocupar d'un problema en qualsevol vèrtex del seu entorn tancat. En general, una entitat pot
consistir en un robot, un observador, una legió, un guàrdia, etc. Informalment, diem que un graf està protegit sota una
determinada ubicació d'entitats si hi ha almenys una entitat disponible per tractar un problema en qualsevol vèrtex.
S'han considerat diverses estratègies (o regles d'ubicació d'entitats), sota cadascuna de les quals el graf es considera
protegit. Aquestes estratègies de protecció de grafs s'emmarquen en la teoria de la dominació en grafs, o en la teoria
de la dominació segura en grafs.
En aquesta tesi, introduïm l'estudi de la w-dominació (segura) en grafs, el qual és un enfocament unificat a la idea de
protecció de grafs, i que engloba variants conegudes de dominació (segura) en grafs i introdueix de noves.
La tesi està estructurada com un compendi de deu articles, els quals han estat publicats en revistes indexades en el
JCR. El primer està dedicat a l'estudi de la w-dominació, el cinquè a l'estudi de la w-dominació segura, mentre que els
altres treballs estan dedicats a casos particulars d'estratègies de protecció total. Com és d'esperar, el nombre mínim
d'entitats necessàries per a la protecció sota cada estratègia és d'interès. En general, s'obtenen fórmules tancades o
fites ajustades sobre els paràmetres estudiats.Supongamos que una o varias entidades están situadas en algunos de los vértices de un grafo simple y que una
entidad situada en un vértice puede ocuparse de un problema en cualquier vértice de su vecindad cerrada. En general,
una entidad puede consistir en un robot, un observador, una legión, un guardia, etc. Informalmente, decimos que un
grafo está protegido bajo una determinada ubicación de entidades si existe al menos una entidad disponible para tratar
un problema en cualquier vértice. Se han considerado varias estrategias (o reglas de ubicación de entidades), bajo
cada una de las cuales el grafo se considera protegido. Estas estrategias de protección de grafos se enmarcan en la
teoría de la dominación en grafos, o en la teoría de la dominación segura en grafos.
En esta tesis, introducimos el estudio de la w-dominación (segura) en grafos, el cual es un enfoque unificado a la idea
de protección de grafos, y que engloba variantes conocidas de dominación (segura) en grafos e introduce otras
nuevas. La tesis está estructurada como un compendio de diez artículos, los cuales han sido publicados en revistas
indexadas en el JCR. El primero está dedicado al estudio de la w-dominación, el quinto al estudio de la w-dominación
segura, mientras que los demás trabajos están dedicados a casos particulares de estrategias de protección total.
Como es de esperar, el número mínimo de entidades necesarias para la protección bajo cada estrategia es de interés.
En general, se obtienen fórmulas cerradas o cotas ajustadas sobre los parámetros estudiadosSuppose that one or more entities are stationed at some of the vertices of a simple graph and that an entity at a vertex
can deal with a problem at any vertex in its closed neighbourhood. In general, an entity could consist of a robot, an
observer, a legion, a guard, and so on. Informally, we say that a graph is protected under a given placement of
entities if there exists at least one entity available to handle a problem at any vertex. Various strategies (or rules for
entities placements) have been considered, under each of which the graph is deemed protected. These strategies for
the protection of graphs are framed within the theory of domination in graphs, or in the theory of secure domination in
graphs.
In this thesis, we introduce the study of (secure) w-domination in graphs, which is a unified approach to the idea of
protection of graphs, that encompasses known variants of (secure) domination in graphs and introduces new ones.
The thesis is structured as a compendium of ten papers which have been published in JCR-indexed journals. The first
one is devoted to the study of w-domination, the fifth one is devoted to the study of secure w-domination, while the
other papers are devoted to particular cases of total protection strategies. As we can expect, the minimum number of
entities required for protection under each strategy is of interest. In general, we obtain closed formulas or tight bounds
on the studied parameters
A Note on Outer-Independent 2-Rainbow Domination in Graphs
Let G be a graph with vertex set V(G) and f:V(G)→{∅,{1},{2},{1,2}} be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: (i)V∅={x∈V(G):f(x)=∅} is an independent set of G. (ii)∪u∈N(v)f(u)={1,2} for every vertex v∈V∅. The outer-independent 2-rainbow domination number of G, denoted by γoir2(G), is the minimum weight ω(f)=∑x∈V(G)|f(x)| among all outer-independent 2-rainbow dominating functions f on G. In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β(G)≤γoir2(G)≤2β(G), where β(G) denotes the vertex cover number of G. Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain
Italian Domination on Ladders and Related Products
An Italian dominating function on a graph is a function such that , and for each vertex for which , we have . The weight of an Italian dominating function is . The minimum weight of all such functions on a graph is called the Italian domination number of . In this thesis, we will consider Italian domination in various types of products of a graph with the complete graph . We will find the value of the Italian domination number for ladders, specific families of prisms, mobius ladders and related products including categorical products and lexicographic products . Finally, we will conclude with open problems
On the {2}-domination number of graphs
[EN] Let G be a nontrivial graph and k ¿ 1 an integer. Given a vector of nonnegative integers
w = (w0,...,wk), a function f : V(G) ¿ {0,..., k} is a w-dominating function on G if f(N(v)) ¿ wi
for every v ¿ V(G) such that f(v) = i. The w-domination number of G, denoted by ¿w(G), is the
minimum weight ¿(f) = ¿v¿V(G)
f(v) among all w-dominating functions on G. In particular, the {2}-
domination number of a graph G is defined as ¿{2}
(G) = ¿(2,1,0)
(G). In this paper we continue with
the study of the {2}-domination number of graphs. In particular, we obtain new tight bounds on this
parameter and provide closed formulas for some specific families of graphs.Cabrera-Martínez, A.; Conchado Peiró, A. (2022). On the {2}-domination number of graphs. AIMS Mathematics. 7(6):10731-10743. https://doi.org/10.3934/math.202259910731107437
Outer Independent Double Italian Domination of Some Graph Products
An outer independent double Italian dominating function on a graph is a function for which each vertex with then and vertices assigned under are independent. The outer independent double Italian domination number is the minimum weight of an outer independent double Italian dominating function of graph . In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, lexicographic product and direct product of custom graphs in terms of this parameter. We also provide the best possible upper and lower bounds for these three products for arbitrary graphs
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