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

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

Some new results on the k-tuple domination number of graphs

Let k ≥ 1 be an integer and G be a graph of minimum degree δ(G) ≥ k − 1. A set D ⊆ V(G) is said to be a k-tuple dominating set of G if |N[v] ∩ D| ≥ k for every vertex v ∈ V(G), where N[v] represents the closed neighbourhood of vertex v. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this paper, we continue with the study of this classical domination parameter in graphs. In particular, we provide some relationships that exist between the k-tuple domination number and other classical parameters, like the multiple domination parameters, the independence number, the diameter, the order and the maximum degree. Also, we show some classes of graphs for which these relationships are achieved

A new lower bound for the independent domination number of a tree

A set D of vertices in a graph G is an independent dominating set of G if D is an independent set and every vertex not in D is adjacent to a vertex in D. The independent domination number of G, denoted by i(G), is the minimum cardinality among all independent dominating sets of G. In this paper we show that if T is a nontrivial tree, then i(T) ≥ n(T)+γ(T)−l(T)+2/4, where n(T), γ(T) and l(T) represent the order, the domination number and the number of leaves of T, respectively. In addition, we characterize the trees achieving this new lower bound

Real patterns and indispensability

While scientific inquiry crucially relies on the extraction of patterns from data, we still have a very imperfect understanding of the metaphysics of patterns—and, in particular, of what it is that makes a pattern real. In this paper we derive a criterion of real-patternhood from the notion of conditional Kolmogorov complexity. The resulting account belongs in the philosophical tradition, initiated by Dennett (1991), that links real-patternhood to data compressibility, but is simpler and formally more perspicuous than other proposals defended heretofore in the literature. It also successfully enforces a non-redundancy principle, suggested by Ladyman and Ross (2007), that aims at excluding as real those patterns that can be ignored without loss of information about the target dataset, and which their own account fails to enforce

índice de sítio diamétrico: um método alternativo para estimar a qualidade do sítio em florestas de Nothofagus obliqua E N. alpina

The first step for constructing models of tree growth and yield is site quality assessment. To estimate this attribute, several methodologies are available in which site index (SI) is a standard one. However, this approach, that uses height at a reference age of trees, can be simplified if age is replaced by another reference variable easier to measure. In this case, the diametric site index (DSI) represents the mean height of dominant trees at a reference mean diameter at breast height. The aim of this work was to develop DSI in pure and mixed Nothofagus alpina and N. obliqua forests, and compare these models with the classical proposals based on height-age variables, within the temperate forest of northwestern Patagonia from Argentina, South America. Data originated from temporary plots and stem analyses were used. Tree age and diameter at breast height were obtained from each plot and used for establishing DSI family functions, following the guide-curve methodology. Site classes were proportionally represented among DSI curves of 17.0, 21.5, 26.0, 30.5 and 35.0 m of dominant tree height. Reference diameter instead of reference age can be cautiously used in order to fit site index models.Primeiro passo para a construção de modelos de crescimento e produção de árvores e a avaliação da qualidade do sítio. Para estimar este atributo, várias metodologias estão disponíveis, na qual o índice de sítio (IS) é padrão. No entanto, esta abordagem, que utiliza uma altura na idade de referência, pode ser simplificada se a idade é substituída por outra variável de referência mais fácil de medir. Neste caso, o índice de índice de sítio diamétrico (ISD) representa a altura média das árvores dominantes de um diâmetro à altura do peito referência. O objetivo deste trabalho foi desenvolver ISD para florestas puras e mistas de Nothofagus alpina e N. obliqua, e comparar esses modelos com as propostas clássicas baseadas nas variáveis altura-idade, para a floresta temperada do noroeste da Patagônia da Argentina, América do Sul. Dados provenientes de parcelas temporárias e análises de tronco foram utilizados. Foram obtidos idade e diâmetro à altura do peito de cada parcela e utilizados para o estabelecimento das funções da família DSI, seguindo a metodologia da curva-guia. Classes de sítio foram proporcionalmente representados entre curvas DSI de 17,0; 21,5; 26,0; 30,5 e 35,0 m de altura da árvore dominante. O diâmetro de referência em vez da idade de referência pode ser usado com cautela para ajustar modelos de índice de sítio.Fil: Attis Beltran, Hernan. Universidad Nacional del Comahue. Asentamiento Universidad San Martin de Los Andes; Argentina. Universidad Nacional del Comahue; ArgentinaFil: Chauchards, Luis Mario. Universidad Nacional del Comahue; ArgentinaFil: Velásquez, Abel. Universidad Nacional del Comahue; ArgentinaFil: Sbrancia, Renato. Universidad Nacional del Comahue; ArgentinaFil: Martínez Pastur, Guillermo José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Austral de Investigaciones Científicas; Argentina. Universidad Nacional del Comahue; Argentin

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

Relating the super domination and 2-domination numbers in cactus graphs

A set D⊆V(G) is a super dominating set of a graph G if for every vertex u∈V(G)\D , there exists a vertex v∈D such that N(v)\D={u} . The super domination number of G , denoted by γsp(G) , is the minimum cardinality among all super dominating sets of G . In this article, we show that if G is a cactus graph with k(G) cycles, then γsp(G)≤γ2(G)+k(G) , where γ2(G) is the 2-domination number of G . In addition, and as a consequence of the previous relationship, we show that if T is a tree of order at least three, then γsp(T)≤α(T)+s(T)−1 and characterize the trees attaining this bound, where α(T) and s(T) are the independence number and the number of support vertices of T , respectively