Existence of pairwise internally disjoint cut segments

Determine whether every rooted dual spanning tree of a coronoid system admits n pairwise internally disjoint cut segments whose branch vertices correspond only to non-hexagonal faces, where n is the number of holes.

Background

The paper’s flow criterion for determining connected components of the resonance graph of a coronoid system uses n chosen cut segments in a rooted tree of the dual graph, with one segment associated to each hole. The authors observe that, in a potentially stronger formulation, these segments could be required to be pairwise internally disjoint and assigned arbitrary orientations.

The authors explain that only minor changes to the proof would be needed if such a family always existed. They explicitly leave unresolved whether every coronoid system has a rooted dual tree containing such a family of disjoint cut segments.