Eulerian k-dominating reconfiguration graphs

For a graph G, the vertices of the k-dominating graph, denoted Dₖ(G), correspond to the dominating sets of G with cardinality at most k. Two vertices of Dₖ(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 Dₖ(G) is not necessarily connected when $k < |V(G)|$, much research has focused on conditions under which Dₖ(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 $k = |V(G)|$, we characterize those graphs G for which Dₖ(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.