The Algorithmic Complexity of Domination Digraphs

Let G = (V,E) be an undirected graph and let π = {V1, V2, . . . , Vk} be a partition of the vertices V of G into k blocks Vi. From this partition one can construct the following digraph D(π) = (π,E(π)), the vertices of which correspond one-to-one with the k blocks Vi of π, and there is an arc from Vi to Vj if every vertex in Vj is adjacent to at least one vertex in Vi, that is, Vi dominates Vj . We call the digraph D(π) the domination digraph of π. A triad is one of the 16 digraphs on three vertices having no loops or multiple arcs. In this paper we study the algorithmic complexity of deciding if an arbitrary graph G has a given digraph as one of its domination digraphs, and in particular, deciding if a given triad is one of its domination digraphs. This generalizes results for the domatic number.