Given a connected $d$-regular graph $G=(V,E)$, generate a sequence of minors by performing *only* edge contractions and loop deletions (as, e.g., in Karger's algorithm) until the graph collapses to a single vertex. I am interested in protocols for which one can obtain (asymptotic) upper bounds on the *maximum* degree across all minors generated as a function of $|V|$.

A naive contraction sequence and bound: pick a vertex and keep contracting its incident edges. Then the maximum degree encountered during the process will scale as $|V|$, unless $G$ is a cycle.

Intuition (and some numerical experiments) suggest that there must be algorithms and / or other subclasses of graphs (i.e., planar, small $d$, or others) for which the scaling is slower (possibly sublinear in $|V|$). For example, choosing an edge contraction sequence according to the minimum cut of $G$ and each subsequent minor seems to always be scaling better than the naive scheme.

It seems to me that this problem must have been studied before. Are there any examples in the literature?