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

The number of independent sets in bipartite graphs and benzenoids

Given a graph $G$, we study the number of independent sets in $G$, denoted $i(G)$. This parameter is known as both the Merrifield-Simmons index of a graph as well as the Fibonacci number of a graph. In this paper, we give general bounds for $i(G)$ when $G$ is bipartite and we give its exact value when $G$ is a balanced caterpillar. We improve upon a known upper bound for $i(T)$ when $T$ is a tree, and study a conjecture that all but finitely many positive integers represent $i(T)$ for some tree $T$. We also give exact values for $i(G)$ when $G$ is a particular type of benzenoid

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]

Graphs with equal Grundy domination and independence number

The Grundy domination number, ${gamma_{rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,ldots, v_k)$ of vertices in $G$ such that for every $iin {2,ldots, k}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $Gamma(G)$, which is in turn greater than or equal to the independence number $alpha(G)$. In this paper, we initiate the study of the class of graphs $G$ with $Gamma(G)={gamma_{rm gr}}(G)$ and its subclass consisting of graphs $G$ with $alpha(G)={gamma_{rm gr}}(G)$. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs $G$ with $Gamma(G)={gamma_{rm gr}}(G)$ and present large families of such graphs

On Well-Covered Direct Products

A graph G is well-covered if all maximal independent sets of G have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well-covered graphs with independence number one-half their order is well-covered. We initiate a characterization of nontrivial connected well-covered graphs G and H, whose independence numbers are strictly less than one-half their orders, such that their direct product G × H is well-covered. In particular, we show that in this case both G and H have girth 3 and we present several infinite families of such well-covered direct products. Moreover, we show that if G is a factor of any well-covered direct product, then G is a complete graph unless it is possible to create an isolated vertex by removing the closed neighborhood of some independent set of vertices in G

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