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

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

Roman domination in direct product graphs and rooted product graphs1

Let G be a graph with vertex set V(G). A function f : V(G) -> {0, 1, 2) is a Roman dominating function on G if every vertex v is an element of V(G) for which f(v) = 0 is adjacent to at least one vertex u is an element of V(G) such that f(u) = 2. The Roman domination number of G is the minimum weight omega(f) = Sigma(x is an element of V(G)) f(x) among all Roman dominating functions f on G. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination number, and the Roman domination number of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.The second author (Iztok Peterin) has been partially supported by the Slovenian Research Agency by the projects No. J1-1693 and J1-9109. The last author (Ismael G. Yero) has been partially supported by "Junta de Andalucia", FEDER-UPO Research and Development Call, reference number UPO1263769

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

Double domination in lexicographic product graphs

[EN] In a graph G, a vertex dominates itself and its neighbours. A subset S subset of V(G) is said to be a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality among all double dominating sets of G is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs G o H in terms of invariants of the factor graphs G and H.Cabrera Martínez, A.; Cabrera García, S.; Rodríguez-Velázquez, J. (2020). Double domination in lexicographic product graphs. Discrete Applied Mathematics. 284:290-300. https://doi.org/10.1016/j.dam.2020.03.045S290300284Ahangar Abdollahzadeh, H., Henning, M., Samodivkin, V., & Yero, I. (2016). Total Roman domination in graphs. Applicable Analysis and Discrete Mathematics, 10(2), 501-517. doi:10.2298/aadm160802017aAmjadi, J., Sheikholeslami, S. M., & Soroudi, M. (2019). On the total Roman domination in trees. Discussiones Mathematicae Graph Theory, 39(2), 519. doi:10.7151/dmgt.2099Cabrera Martínez, A., Cabrera García, S., & Carrión García, A. (2020). Further Results on the Total Roman Domination in Graphs. Mathematics, 8(3), 349. doi:10.3390/math8030349A. Cabrera Martínez, J.A. Rodríguez-Velázquez, Total protection of lexicographic product graphs, Discuss. Math. Graph Theory, in press.Campanelli, N., & Kuziak, D. (2019). Total Roman domination in the lexicographic product of graphs. Discrete Applied Mathematics, 263, 88-95. doi:10.1016/j.dam.2018.06.008Cockayne, E. J., Dawes, R. M., & Hedetniemi, S. T. (1980). Total domination in graphs. Networks, 10(3), 211-219. doi:10.1002/net.3230100304Dettlaff, M., Lemańska, M., Rodríguez-Velázquez, J. A., & Zuazua, R. (2019). On the super domination number of lexicographic product graphs. Discrete Applied Mathematics, 263, 118-129. doi:10.1016/j.dam.2018.03.082Dorbec, P., Mollard, M., Klavžar, S., & Špacapan, S. (2008). Power Domination in Product Graphs. SIAM Journal on Discrete Mathematics, 22(2), 554-567. doi:10.1137/060661879Hajian, M., & Rad, N. J. (2019). A new lower bound on the double domination number of a graph. Discrete Applied Mathematics, 254, 280-282. doi:10.1016/j.dam.2018.06.009Harant, J., & Henning, M. . A. (2005). On Double Domination in Graphs. Discussiones Mathematicae Graph Theory, 25(1-2), 29. doi:10.7151/dmgt.1256Chellali, M., & Khelifi, S. (2012). Double domination critical and stable graphs upon vertex removal. Discussiones Mathematicae Graph Theory, 32(4), 643. doi:10.7151/dmgt.1633Liu, C.-H., & Chang, G. J. (2012). Roman domination on strongly chordal graphs. Journal of Combinatorial Optimization, 26(3), 608-619. doi:10.1007/s10878-012-9482-yNowakowski, R. J., & Rall, D. F. (1996). Associative graph products and their independence, domination and coloring numbers. Discussiones Mathematicae Graph Theory, 16(1), 53. doi:10.7151/dmgt.1023Šumenjak, T. K., Rall, D. F., & Tepeh, A. (2013). Rainbow domination in the lexicographic product of graphs. Discrete Applied Mathematics, 161(13-14), 2133-2141. doi:10.1016/j.dam.2013.03.011Šumenjak, T. K., Pavlič, P., & Tepeh, A. (2012). On the Roman domination in the lexicographic product of graphs. Discrete Applied Mathematics, 160(13-14), 2030-2036. doi:10.1016/j.dam.2012.04.008Valveny, M., Pérez-Rosés, H., & Rodríguez-Velázquez, J. A. (2019). On the weak Roman domination number of lexicographic product graphs. Discrete Applied Mathematics, 263, 257-270. doi:10.1016/j.dam.2018.03.03

Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number