Strong Equality of Perfect Roman and Weak Roman Domination in Trees

Let G=(V,E) be a graph and f:V⟶{0,1,2} be a function. Given a vertex u with f(u)=0, if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with f(u)=0 has a neighbor v with f(v)>0 and the function f′:V⟶{0,1,2} with f′(u)=1, f′(v)=f(v)−1, f′(w)=f(w) if w∈V∖{u,v} has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with f(u)=0 is adjacent to exactly one vertex v for which f(v)=2. Let the weight of f be w(f)=∑v∈Vf(v). The weak (resp., perfect) Roman domination number, denoted by γr(G) (resp., γpR(G)), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating

Further Results on the Total Roman Domination in Graphs

[EN] Let G be a graph without isolated vertices. A function f:V(G)-> {0,1,2} is a total 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 , and if the subgraph induced by the set {v is an element of V(G):f(v)>= 1} has no isolated vertices. The total Roman domination number of G, denoted gamma tR(G) , is the minimum weight omega (f)=Sigma v is an element of V(G)f(v) among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for gamma tR(G) which improve the well-known bounds 2 gamma (G)<= gamma tR(G)<= 3 gamma (G) , where gamma (G) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.Cabrera Martínez, A.; Cabrera García, S.; Carrión García, A. (2020). Further Results on the Total Roman Domination in Graphs. Mathematics. 8(3):1-8. https://doi.org/10.3390/math8030349S1883Henning, M. A. (2009). A survey of selected recent results on total domination in graphs. Discrete Mathematics, 309(1), 32-63. doi:10.1016/j.disc.2007.12.044Henning, M. A., & Yeo, A. (2013). Total Domination in Graphs. Springer Monographs in Mathematics. doi:10.1007/978-1-4614-6525-6Henning, M. A., & Marcon, A. J. (2016). Semitotal Domination in Claw-Free Cubic Graphs. Annals of Combinatorics, 20(4), 799-813. doi:10.1007/s00026-016-0331-zHenning, M. . A., & Marcon, A. J. (2016). Vertices contained in all or in no minimum semitotal dominating set of a tree. Discussiones Mathematicae Graph Theory, 36(1), 71. doi:10.7151/dmgt.1844Henning, M. A., & Pandey, A. (2019). Algorithmic aspects of semitotal domination in graphs. Theoretical Computer Science, 766, 46-57. doi:10.1016/j.tcs.2018.09.019Cockayne, E. J., Dreyer, P. A., Hedetniemi, S. M., & Hedetniemi, S. T. (2004). Roman domination in graphs. Discrete Mathematics, 278(1-3), 11-22. doi:10.1016/j.disc.2003.06.004Stewart, I. (1999). Defend the Roman Empire! Scientific American, 281(6), 136-138. doi:10.1038/scientificamerican1299-136Chambers, E. W., Kinnersley, B., Prince, N., & West, D. B. (2009). Extremal Problems for Roman Domination. SIAM Journal on Discrete Mathematics, 23(3), 1575-1586. doi:10.1137/070699688Favaron, O., Karami, H., Khoeilar, R., & Sheikholeslami, S. M. (2009). On the Roman domination number of a graph. Discrete Mathematics, 309(10), 3447-3451. doi:10.1016/j.disc.2008.09.043Liu, C.-H., & Chang, G. J. (2012). Upper bounds on Roman domination numbers of graphs. Discrete Mathematics, 312(7), 1386-1391. doi:10.1016/j.disc.2011.12.021González, Y., & Rodríguez-Velázquez, J. (2013). Roman domination in Cartesian product graphs and strong product graphs. Applicable Analysis and Discrete Mathematics, 7(2), 262-274. doi:10.2298/aadm130813017gLiu, 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-yAhangar 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., Montejano, L. P., & Rodríguez-Velázquez, J. A. (2019). Total Weak Roman Domination in Graphs. Symmetry, 11(6), 831. doi:10.3390/sym1106083

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Defending the Roman Empire—A new strategy

AbstractMotivated by an article by Ian Stewart (Defend the Roman Empire!, Scientific American, Dec. 1999, pp. 136–138), we explore a new strategy of defending the Roman Empire that has the potential of saving the Emperor Constantine the Great substantial costs of maintaining legions, while still defending the Roman Empire. In graph theoretic terminology, let G=(V,E) be a graph and let f be a function f:V→{0,1,2}. A vertex u with f(u)=0 is said to be undefended with respect to f if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u)=0 is adjacent to a vertex v with f(v)>0 such that the function f′:V→{0,1,2}, defined by f′(u)=1, f′(v)=f(v)−1 and f′(w)=f(w) if w∈V−{u,v}, has no undefended vertex. The weight of f is w(f)=∑v∈Vf(v). The weak Roman domination number, denoted γr(G), is the minimum weight of a WRDF in G. We show that for every graph G, γ(G)⩽γr(G)⩽2γ(G). We characterize graphs G for which γr(G)=γ(G) and we characterize forests G for which γr(G)=2γ(G)

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