Weighted vertex-cover FPT for Set of Degrees MST

Determine whether Set of Degrees MST is fixed-parameter tractable with respect to the vertex cover number on general edge-weighted graphs.

Background

The paper proves that Set of Degrees MST is fixed-parameter tractable when parameterized by the vertex cover number for unweighted graphs. The weighted case remains unresolved: the authors identify this as a prominent gap in the complexity classification. They indicate that resolving it may depend on obtaining a fixed-parameter algorithm for generalized B-matching on weighted bipartite graphs. The paper establishes tractability for a restricted weighted setting when the vertex-cover number is combined with the maximum number of admissible degrees per vertex, but this does not settle the unrestricted edge-weighted case.

References

Perhaps most prominently, we do not yet know whether Set of Degrees MST is fixed-parameter tractable w.r.t. the vertex cover number in the general setting of edge-weighted graphs; we believe that settling this question hinges on obtaining a fixed-parameter algorithm for generalized $B$-matching on weighted bipartite graphs.

— Not All Degree Constraints Are Created Equal when Computing Spanning Trees  (2608.25530 - Bojikian et al., 26 Aug 2026) in Section 6, Concluding Remarks (Section~\ref{sec:conclusion})