Geodesic
Eulerian $k$-dominating reconfiguration graphs
Discrete mathematics & theoretical computer science, Tome 27 (2025) no. 2.

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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 $k < |V(G)|$, 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 $k = |V(G)|$, 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. 
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     author = {Messinger, M. E. and Porter, A.},
     title = {Eulerian $k$-dominating reconfiguration graphs},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {2025},
     doi = {10.46298/dmtcs.13438},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.13438/}
}
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Messinger, M. E.; Porter, A. Eulerian $k$-dominating reconfiguration graphs. Discrete mathematics & theoretical computer science, Tome 27 (2025) no. 2. doi : 10.46298/dmtcs.13438. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.13438/

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