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 distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r > 2, there exists a connected graph H, such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G,J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

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.

For a partial cube G its crossing graph $G^#$ is the graph whose vertices are the ϴ‐classes of G, two classes being adjacent if they cross on some cycle in G. The following problem posed in [S. Klavžar and H. M. Mulder, SIAM J. Discrete Math., 15 (2002), pp. 235–251, Problem 7.1] is considered: What can be said about the partial cube G if $G^#$ is the join $A\oplus B$ of graphs A and B with at least one edge? It is proved that for arbitrary graphs A and B, where at least one of them contains an edge, there exists a Cartesian prime partial cube G such that $G^# = A\oplus B$. On the other hand, if G is a median graph, then $G^# = A\oplus B$ if and only if $G=H\,\square\, K$, where $H^# = A$ and $K^# = B$. Along the way some new facts about partial cubes are obtained; for instance, a bipartite graph of radius 2 is a partial cube if and only if it is $K_{2,3}$‐free.

Let G be a connected bipartite graph. An involution $\alpha$ of G that preserves the bipartition of G is called bipartite. Let $G^\alpha$ be the graph obtained from G by adding to G the natural perfect matching induced by $\alpha$. We show that the k-cube Q k is isomorphic to the direct product $G \times H$ if and only if G is isomorphic to $Q_{k-1}^\alpha$ for some bipartite involution $alpha$ of $Q_{k-1}$ and H=K 2 .