Characterization of connected mincut graphs

Determine the structural conditions on a graph $G$ that ensure that its mincut graph $X(G)$ is connected, equivalently, that every mincut or subset of mutually intersecting mincuts intersects another mincut or intersecting subset.

Background

The mincut graph $X(G)$ has one vertex for each minimum edge-cut of $G$, with adjacency defined by nonempty intersection of the corresponding edge-cut sets. If $X(G)$ is disconnected, then the next iteration produces the null graph $K_0$.

The paper raises the unresolved problem of identifying graph properties that guarantee connectedness of $X(G)$ and suggests that the proportion of trivial vertices may be relevant, but does not establish a sufficient or necessary characterization.