Roman Domination Cover Rubbling

In this thesis, we introduce Roman domination cover rubbling as an extension of domination cover rubbling. We define a parameter on a graph $G$ called the \textit{Roman domination cover rubbling number}, denoted $\rho_{R}(G)$, as the smallest number of pebbles, so that from any initial configuration of those pebbles on $G$, it is possible to obtain a configuration which is Roman dominating after some sequence of pebbling and rubbling moves. We begin by characterizing graphs $G$ having small $\rho_{R}(G)$ value. Among other things, we also obtain the Roman domination cover rubbling number for paths and give an upper bound for the Roman domination cover rubbling number of a tree

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

Some Properties on Strong Roman Domination in Graphs

A Strong Roman dominating function (SRDF) is a function satisfying the condition that every vertex for which is adjacent to at least one vertex for which and every vertex for which is adjacent to at least one vertex for which . The weight of an SRDF is the value . The minimum weight of an SRDF on a graph is called the Strong Roman domination numberof . In this paper, we attempt to verify some properties on SRDF and moreover we present Strong Roman domination number for some special classes of graphs. Also we show that for a tree with vertices, leaves and support vertices, we have and we characterize all trees achieving this bound

The Roman Legacy on European Chestnut and Walnut Arboriculture

The political and administrative unification process under the Roman Empire resulted not only in a progressive linguistic, religious, and cultural homogenisation of the concerned population, but also in the need of trade and exchanges for satisfying new dietary habits and markets. In this paper, we use palaeoecological pollen and macroremain records extracted from Neotoma online database and from scientific literature to analyse the contribution of the Romans in the distribution and cultivation history of Castanea sativa and Juglans regia, two tree species, which are usually considered highly connected with Romanisation. Our results highlight a substantial difference in the impact of the Romans on the history of the two species between the territories of the Western and Eastern Empire, where the Roman influence is much less evident. In the western territories, Juglans regia experienced a significant pre-Roman increase and spread, which was only partially intensified under Roman domination. Castanea sativa, on the contrary, benefited from a significant boost following the Roman conquest, especially in the mountain areas south of the Alps and in France. After the fall of the Roman Empire this led to the traditional medieval chestnut civilisation in which the chestnut tree became a main source of livelihood