The orientable domination number, DOM(G), of a graph G is the largest domination number over all orientations of G. In this paper, 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. (1996) is extended by establishing the values of DOM(Kn1,n2,n3) for arbitrary positive integers n1,n2 and n3. 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 Brešar et al. (2022).

The domination game played on a graph G consists of two players, Dominator and Staller, who alternate taking turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay the process as much as possible. The game domination number $\gamma_g(G)$ (resp., $\gamma_g'(G)$) is the number of vertices chosen when Dominator (resp., Staller) starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph G, $\gamma(G)\leq\gamma_g(G)\leq2\gamma(G)-1$, and that all possible values can be realized. It is proved that for any graph G, $\gamma_g(G)-1\leq\gamma'_g(G)\leq\gamma_g(G)+2$, and that most of the possibilities for mutual values of $\gamma_g(G)$ and $\gamma_g'(G)$ can be realized. A connection with Vizing's conjecture is established, and a lower bound on the game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

In this paper, we consider a variation of total domination in which we limit the ability of a vertex to dominate its neighbors in one of two ways: (a) Every vertex in the dominating set dominates exactly k of its neighbors. Graphs that have such dominating sets are characterized and a recognition algorithm for trees is described. (b) Every vertex in the dominating set dominates no more than k of its neighbors. It is shown that the existence of such a dominating set is equivalent to the existence in the graph of a star-factor (which is a partition of the vertex set into m-stars where 1 ≤ m ≤ k). It is further shown that the existence of such a star-factor is equivalent to a Tutte-like condition which requires that k|N(I)| ≥ |I| for every independent set I of vertices in G. When this last result is interpreted in the case when G is bipartite, a generalization of the Marriage Theorem emerges. © 1996 John Wiley & Sons, Inc.

Let $\beta (G)$ denote the independence number of a graph G. We introduce $A(G) = \lim_{k \to \infty } \beta (G^k )/| V(G) |^k $, where the categorical graph product is used. This limit, surprisingly, lies in the range $( 0,1/2 ] \cup \{ 1 \}$. We can show that this limit can take any such rational number, but is there any G for which $A(G)$ is irrational? A useful technique for bounding $A(G)$ is to consider special spanning subgraphs. These bounds allow us to efficiently compute $A(G)$ for many G. We give a condition which if true for G shows that $A(G) > \beta (G)/| V(G) |$. This brings up the question; for which G does $A(G) = \beta (G)/| V(G) |$? This happens if G is a Cayley graph of an Abelian group or if G is a connected graph that has an automorphism which has a single orbit.