Graphs with disjoint 2-dominating sets

A subset D ⊆ V (G) is a dominating set of a multigraph G if every vertex in V (G) \ D has a neighbor in D, while D is a 2-dominating set of G if every vertex belonging to V (G) \ D is joined by at least two edges with a vertex or vertices in D. A graph G is a (2, 2)-dominated graph if it has a pair (D, D^′) of disjoint 2-dominating sets of vertices of G. In this paper we present two characterizations of minimal (2, 2)-dominated graphs.