Orientable domination in product-like graphs

The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377]

Himpunan Dominasi Ganda Pada Graf Korona Dan Graf Produk Leksikografi Dua Buah Graf

ABSTARCT. Let be a subset of , with is a graph without isolated vertices. A subset of referred to double domination in if every vertex , such that every vertex in minimal adjacent with two element of . The minimum cardinality of domination set, total domination set, and double domination set in respectively is a is a domination number, total domination number, and double domination number denote respectively , , and . A double domination number in minimum is two and a double domination number in will not be more order () in , that . A domination number if add one vertex of domination in then the element of dimination number will not be more a double domination number in , that . In this final project examined the sum of bound of doubel domination number in Corona and product lexicographic of two graphs. The minimum cardinality of double domination in Corona is with is order in . Menwhile, the minimum cardinality of double domination in Product Lexicographic at most

Total Roman domination in the lexicographic product of graphs

A total Roman dominating function of a graph $G=(V,E)$ is a function $f: V(G)\to \{0,1,2\}$ such that for every vertex $v$ with $f(v)=0$ there exists a vertex $u$ adjacent to $v$ with $f(u)=2$, and such that the subgraph induced by the set of vertices labeled one or two has no isolated vertices. The total Roman domination number of $G$ is the minimum value of the sums $\sum_{v\in V}{f(v)}$ over all total Roman dominating functions $f$ of $G$. In this paper we study the total Roman domination number of the lexicographic product of graphs

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