The separability characterization of matching-stable graphs

Let $G$ be a connected non-bipartite graph. A matching queue $(G,\lambda,\Phi)_{scriptsize{\textsc{c}}}$ has arrival-rate vector $\lambda$ in $textsc{Ncond}_{scriptsize{\textsc{c}}}(G)$, where $\Phi$ is an admissible matching policy. A graph is matching-stable if the corresponding matching queue is stable for every admissible matching policy and every arrival-rate vector satisfying this condition. A graph is separable if it has the separability property used in the paper, and its order is the number of vertices in the relevant separable component.

Separability characterization conjecture. The only connected and non-bipartite graphs $G$ for which the matching queue $(G,\lambda,\Phi)_{\scriptsize{\textsc{c}}}$ is stable for any admissible matching policy $\Phi$ and any $\lambda \in \textsc{Ncond}_{\scriptsize{\textsc{c}}}(G)$ are the separable graphs of order $3$ or more.

The conjecture would extend the paper's characterization from the complement of the class of graphs inducing an odd cycle of size at least $7$ without a $5$-cycle or pendant graph. Equivalently, since graphs in that remaining class are non-separable, it asserts that no such graph is matching-stable; the paper reports that this remains unproved and identifies instability for the pendant graph and the $5$-cycle as supporting results.