Quasi-total Roman Domination in Graphs

[EN] A quasi-total Roman dominating function on a graph G=(V,E) is a function f:V ->{0,1,2}satisfying the following: Every vertex for which u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2, and If x is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then f(x) = 1. The weight of a quasi-total Roman dominating function is the value omega(f) = f(V) = Sigma(u is an element of V) f(u). The minimum weight of a quasi-total Roman dominating function on a graph G is called the quasi-total Roman domination number of G. We introduce the quasi-total Roman domination number of graphs in this article, and begin the study of its combinatorial and computational properties.Cabrera García, S.; Cabrera Martínez, A.; Yero, IG. (2019). Quasi-total Roman Domination in Graphs. Results in Mathematics. 74(4):1-18. https://doi.org/10.1007/s00025-019-1097-5S11874

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

Dominating the Direct Product of Two Graphs through Total Roman Strategies

Given a graphGwithout isolated vertices, a total Roman dominating function forGis a function f:V(G)->{0,1,2}such that every vertexuwithf(u)=0is adjacent to a vertexvwithf(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number gamma tR(G)ofGis the smallest possible value of n-ary sumation v is an element of V(G)f(v)among all total Roman dominating functionsf. The total Roman domination number of the direct productGxHof the graphsGandHis studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between gamma tR(GxH)and some classical domination parameters for the factors are given. Characterizations of the direct product graphsGxHachieving small values (<= 7) for gamma tR(GxH)are presented, and exact values for gamma tR(GxH)are deduced, while considering various specific direct product classes

RESTRAINED ROMAN REINFORCEMENT NUMBER IN GRAPHS

A restrained Roman dominating function (RRD-function) on a graph $G=(V,E)$ is a function $f$ from $V$ into $\{0,1,2\}$ satisfying: (i)  every vertex $u$ with $f(u)=0$ is adjacent to a vertex $v$ with $f(v)=2$; (ii) the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an RRD-function is the sum of its function value over the whole set of vertices, and the restrained Roman domination number is the minimum weight of an RRD-function on $G.$ In this paper, we begin the study of the restrained Roman reinforcement number $r_{rR}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges that we must add to the graph to decrease its restrained Roman domination number. We first show that the decision problem associated with the restrained Roman reinforcement problem is NP-hard. Then several properties as well as some sharp bounds of the restrained Roman reinforcement number are presented. In particular it is established that $r_{rR}(T)=1$ for every tree $T$ of order at least three

Theoretical Computer Science and Discrete Mathematics

This book includes 15 articles published in the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity

On the Quasi-Total Roman Domination Number of Graphs

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, computer and social networks, monitoring communication, coding theory, and algorithm design, among others. In the last two decades, the functions defined on graphs have attracted the attention of several researchers. The Roman-dominating functions and their variants are one of the main attractions. This paper is a contribution to the Roman domination theory in graphs. In particular, we provide some interesting properties and relationships between one of its variants: the quasi-total Roman domination in graphs

Quasi-total Roman reinforcement in graphs

AbstractA quasi-total Roman dominating function (QTRD-function) on [Formula: see text] is a function [Formula: see text] such that (i) every vertex x for which f(x) = 0 is adjacent to at least one vertex v for which f(v) = 2, and (ii) if x is an isolated vertex in the subgraph induced by the set of vertices with non-zero values, then f(x) = 1. The weight of a QTRD-function is the sum of its function values over the whole set of vertices, and the quasi-total Roman domination number is the minimum weight of a QTRD-function on G. The quasi-total Roman reinforcement number [Formula: see text] of a graph G is the minimum number of edges that have to be added to G in order to decrease the quasi-total Roman domination number. In this paper, we initiate the study of quasi-total Roman reinforcement in graphs. We first show that the decision problem associated with the quasi-total Roman reinforcement problem is NP-hard even when restricted to bipartite graphs. Then basic properties of the quasi-total Roman reinforcement number are provided. Finally, some sharp bounds for [Formula: see text] are also presented