We consider finite graphs G with vertex set V (G). A subset D ⊆ V (G) is a dominating set of the graph G, if every vertex v ∈ V (G) − D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality among the dominating sets of G. In this note, we characterize the trees T with an even number of vertices such that γ(T ) = |V (T )| − 2

Denote the total domination number of a graph G by γt(G). A graph G is said to be total domination edge critical, or simply γtcritical, if γt(G + e) < γt(G) for each edge e ∈ E(G). For 3t-critical graphs G, that is, γt-critical graphs with γt(G) = 3, the diameter of G is either 2 or 3. We characterise the 3t-critical graphs G with diam G = 3.

We show how to count all minimumweighted dominating sets of a graph on n vertices in timeO(1.5535).

Using hypergraph transversals it is proved that γt(Qn+1) = 2γ(Qn), where γt(G) and γ(G) denote the total domination number and the domination number of G, respectively, and Qn is the n-dimensional hypercube. More generally, it is shown that if G is a bipartite graph, then γt(G K2) = 2γ(G). Further, we show that the bipartiteness condition is essential by constructing, for any k > 1, a (non-bipa...

Recently, Azarija et al. considered the prism G K2 of a graph G and showed that γt(G K2) = 2γ(G) if G is bipartite, where γt(G) and γ(G) are the total domination number and the domination number of G. In this note, we give a simple proof and observe that there are similar results for other pairs of parameters. We also answer a question from that paper and show that for all graphs γt(G K2) ≥ 4 3...

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s, s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of...

Let V1, V2 be a partition of the vertex set in a graph G, and let γi denote the least number of vertices needed in G to dominate Vi. We prove that γ1 + γ2 ≤ 45 |V (G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ1 + γ2 for graphs with minimum valenc...

For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the e+1-th blowup dominates the e-th, locally or globally. It is shown that the answer is affirmative (globally for any e) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowup...