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.