Connected k-Dominating Graphs

For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ 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 by $d_{0}(G)$ the smallest integer for which $D_{k}(G)$ is connected for all k greater than or equal to $d_{0}(G)$. It is known that $d_{0}(G)$ lies between $\Gamma(G)+1$ and $|V|$ (inclusive), where ${\Gamma}(G)$ is the upper domination number of G, but constructing a graph G such that $d_{0}(G)>{\Gamma}(G)+1$ appears to be difficult. We present two related constructions. The first construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $G_{k,r}$ such that ${\Gamma}(G_{k,r})=k, {\gamma}(G_{k,r})=r+1$ and $d_{0}(G_{k,r})=k+r={\Gamma}(G)+{\gamma}(G)-1$. The second construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $Q_{k,r}$ such that ${\Gamma}(Q_{k,r})=k, {\gamma}(Q_{k,r})=r$ and $d_{0}(Q_{k,r})=k+r={\Gamma}(G)+{\gamma}(G)$.