The domatic number of block-cactus graphs Original Research Article

A subset D of the vertex set of a graph G is called a dominating set if each vertex of G is either in D or adjacent to some vertex in D. The domatic number d(G) is defined as the maximum cardinality of a partition of the vertex set of G into dominating sets. A graph G for which d(G)=δ(G)+1, where δ(G) denotes the minimum degree of G, is called ‘domatically full’. A graph whose blocks are either cycles or complete is called a ‘block-cactus graph’. In this paper we characterize the domatically full block-cactus graphs and determine the domatic number of all not domatically full graphs of this class, which extends a result of .