Structural characterization of forbidden minors for integral minimum cycle bases

Establish whether the finite family of forbidden minors characterizing graphs whose minimum cycle bases are not always integral admits a useful structural characterization.

Background

The paper defines an \mcbeq{} graph as one for which every edge-weight assignment has equal minimum directed and minimum integral cycle-basis weights. It proves that the class of \mcbeq{} graphs is minor-closed, so a finite set of minimal non-\mcbeq{} graphs, denoted \forbiddenminors, exists. The paper develops algorithms and identifies examples, including forbidden minors related to generalized Petersen graphs, but does not determine the family completely. The authors explicitly identify a useful structural description of this forbidden-minor family as the primary open problem raised by the work.

References

Beyond complete graphs, \forbiddenminors is far from fully determined: the search seeded at $P_{7,2}$ exhibits several of its members, but no complete description is known, and it remains open whether \forbiddenminors admits a useful structural characterization.

— Characterizations and Complexity of Minimum Forward and Integer Cycle Bases  (2609.02317 - Riccardi et al., 2 Sep 2026) in Section 6, Conclusion and future directions