Eulerian $k$-dominating reconfiguration graphs

TL;DR

The paper investigates when Eulerian reconfiguration graphs arise from dominating sets by studying $\mathcal{D}_k(G)$ and $\mathcal{D}(G)$. It provides a complete characterization for the unrestricted case: $\mathcal{D}(G)$ is Eulerian iff $|V(G)|$ is even and $G$ is a cocktail party graph, with a reduction to the disconnected case via Cartesian products. For the restricted case ($\gamma(G)<k<|V(G)|$), it gives explicit Eulerian criteria for many graph families (paths, cycles, complete bipartite graphs, complete graphs, cocktail party graphs, and corona/well-dominated graphs) and relates these to well-dominated graph structure. The results reveal parity-based constraints as a central theme and offer both exact classifications and partial generalizations, while highlighting open questions about broader graph classes and the role of well-dominated graphs in this Eulerian landscape.

Abstract

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.

Figures (5)

• Figure 1: An example of $k$-dominating graphs of $P_4$.
• Figure 2: A visualization of some of the adjacencies (and non-adjacencies) in $G$.
• Figure 3: Dominating sets on paths and cycles with $n \equiv 0,1$ (mod $3$).
• Figure 4: Dominating sets on $P_7$.
• Figure 5: Sets of cardinality $4$ on $C_7$ that contain $v_0$ and $v_1$.

Theorems & Definitions (17)

• Theorem 3
• Corollary 5
• Lemma 6
• proof
• Theorem 7
• proof
• Theorem 8
• proof
• Theorem 9
• proof