Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine

International Journal of Mathematical Combinatorics, Vol.6A

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Perfect Roman Domination and Unique Response Roman Domination

The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Edge-Vertex Dominating Set in Unit Disk Graphs

Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set (referred to as ev-dominating set) of $G$ if every vertex of $G$ is ev-dominated by at least one edge of $S^{ev}$. The minimum cardinality of an ev-dominating set is the ev-domination number. The edge-vertex dominating set problem is to find a minimum ev-domination number. In this paper we prove that the ev-dominating set problem is {\tt NP-hard} on unit disk graphs. We also prove that this problem admits a polynomial-time approximation scheme on unit disk graphs. Finally, we give a simple 5-factor linear-time approximation algorithm

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