A characterization of well-dominated Cartesian products

A graph is well-dominated if all its minimal dominating sets have the same cardinality. In this paper we prove that at least one factor of every connected, well-dominated Cartesian product is a complete graph, which then allows us to give a complete characterization of the connected, well-dominated Cartesian products if both factors have order at least $2$. In particular, we show that $G\,\Box\,H$ is well-dominated if and only if $G\,\Box\,H = P_3 \,\Box\,K_3$ or $G\,\Box\,H= K_n \,\Box\,K_n$ for some $n\ge 2$.Comment: 17 page

Orientable total domination in graphs

Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality among all total dominating sets of $D$. Given an undirected graph $G$, we study the maximum and minimum total domination numbers among all orientations of $G$. That is, we study the upper (or lower) orientable domination number of $G$, $\rm{DOM}_t(G)$ (or $\rm{dom}_t(G)$), which is the largest (or smallest) total domination number over all orientations of $G$. We characterize those graphs with $\rm{DOM}_t(G) =\rm{dom}_t(G)$ when the girth is at least $7$ as well as those graphs with $\rm{dom}_t(G) = |V(G)|-1$. We also consider how these parameters are effected by removing a vertex from $G$, give exact values of $\rm{DOM}_t(K_{m,n})$ and $\rm{dom}_t(K_{m,n})$ and bound these parameters when $G$ is a grid graph

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]

Orientabilna dominacija v grafih produktnega tipa

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].Orientabilno dominantno število, ▫${rm DOM}(G)$▫, grafa ▫$G$▫ je največje dominantno število poljubne orientacije grafa ▫$G$▫. V tem članku število ▫${rm DOM}(G)$▫ raziskujemo na različnih produktih grafov ter ob uporabi različnih operacij nad grafi. Za poljubno korono dveh grafov natanko določimo njeno orientabilno dominantno število, medtem ko za kartezični in leksikografski produkt grafov določimo ostre spodnje in zgornje meje. Rezultat Chartranda in soavtorjev (1996) razširimo tako, da določimo vrednosti ▫${rm DOM}(K_{n_1,n_2,n_3})$▫, kjer so ▫$n_1, n_2$▫ in ▫$n_3$▫ poljubna naravna števila. Ob obravnavi orientabilnega dominantnega števila leksikografskih produktov grafov pridemo tudi do negativnega odgovor na vprašanje iz članka Brešarja in soavtorjev (2022), ki se nanaša na dominantno in pakirno število acikličnih usmerjenih grafov

A New Framework to Approach Vizing’s Conjecture

We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing’s conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: γ (X□Y) ≥ max γ(X□Y)≥max{12γ(X)γt(Y),12γt(X)γ(Y)}\gamma \left( {X \square Y} \right) \ge \max \left\{ {{1 \over 2}\gamma (X){\gamma _t}(Y),{1 \over 2}{\gamma _t}(X)\gamma (Y)} \right\}, where γ stands for the domination number, γt is the total domination number, and X□Y is the Cartesian product of graphs X and Y