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

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Semipaired Domination in Some Subclasses of Chordal Graphs

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$

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