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:where γ stands for the domination number, γ t is the total domination number, and X 2 Y is the Cartesian product of graphs X and Y .

A dominating (respectively, total dominating) set S of a digraph D is a set of vertices in D such that the union of the closed (respectively, open) out-neighborhoods of vertices in S equals the vertex set of D. The minimum 1 2 γ H ( ( ) + 1). This inequality is sharp as demonstrated by an infinite family of examples. Ditrees T and

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.

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph G is also equimatchable, meaning that every maximal matching in G is maximum. In this paper, we show that if G is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then G is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product G H is well-edge-dominated, where G and H have order at least 2, if and only if

Motivated by the concept of well-covered graphs, we define a graph to be wellbicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with wellcovered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered. arXiv:1909.07503v1 [math.CO]

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