Cut - Vertices and Domination In

The paper studies the domatic numbers and the total domatic numbers of graphs having cut-vertices. We shall study the domatic number d(G) and the total domatic number d t (G) of a graph G. A survey of the related theory is given in 3]. We consider nite, undirected graphs without loops or multiple edges. A subset D of the vertex set V (G) of a graph G is called dominating (total dominating), if for each x ∈ V (G) − D (for each x ∈ V (G), respectively) there exists a vertex y ∈ D adjacent to x. A partition D of V (G) is called a domatic (total domatic) partition of G, if each class of D is a dominating (total dominating, respectively) set. The maximum number of classes of a domatic (total domatic) partition of V (G) was in 1] ((2]) named the domatic (total domatic, respectively) number of G, and it is denoted by d(G) (d t (G), respectively). Note that d(G) is well-deened for every nite, undirected graph, while d t (G) is deened only for graphs without isolated vertices. Consider in G a vertex v of minimum valency δ(G). Then a dominating set must contain v or a neighbour of v, thus it is obvious that d(G) 6 δ(G) + 1. A total dominating set must contain a neighbour of v, thus d t (G) 6 δ(G). We shall consider the case when a graph G is the union of two graphs G 1 , G 2 having exactly one common vertex a; this vertex a is a cut-vertex of G. The graphs obtained from G 1 and G 2 by deleting a will be denoted respectively by G 0 1 , G 0 2 .