Eulerian $k$-dominating reconfiguration graphs

For a graph $G$, the vertices of the $k$-dominating graph, denoted ${\mathcal{D}}_k(G)$, correspond to the dominating sets of $G$ with cardinality at most $k$. Two vertices of ${\mathcal{D}}_k(G)$ are adjacent if and only if the corresponding dominating sets in $G$ can be obtained from one other by adding or removing a single vertex of $G$. Since ${\mathcal{D}}_k(G)$ is not necessarily connected when $k<\mathrm{\mid }V(G)\mathrm{\mid }$, much research has focused on conditions under which ${\mathcal{D}}_k(G)$ is connected and recent work has explored the existence of Hamilton paths in the $k$-dominating graph. We consider the complementary problem of determining the conditions under which the $k$-dominating graph is Eulerian. In the case where $k=\mathrm{\mid }V(G)\mathrm{\mid }$, we characterize those graphs $G$ for which ${\mathcal{D}}_k(G)$ is Eulerian. In the case where $k$ is restricted, we determine for a number of graph classes, the conditions under which the $k$-dominating graph is Eulerian.