Connected k-dominating graphs.

For a graph G=(V,E), the k-dominating graph Dk(G) of G has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of Dk(G) are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of G by γ(G) and Γ(G), respectively, and the smallest integer ε for which Dk(G) is connected for all k≥ε by d0(G). It is known that Γ(G)+1≤d0(G)≤|V|, but constructing a graph G such that d0(G)>Γ(G)+1 appears to be difficult.