Combinatorial formulation of the graph enclosure algorithm

Develop a purely combinatorial formulation of the fixed‑parameter tractable algorithm for Graph‑Enclosure‑with‑Penalties (on a connected plane graph with positive edge weights and face penalties) that avoids reliance on a straight‑line geometric embedding and depends only on the graph’s combinatorial embedding (rotation system) and face structure.

Background

To solve Graph‑Enclosure‑with‑Penalties, the paper reduces to a geometric instance via a straight‑line embedding of the input plane graph and then applies a geometric dynamic program. The subproblems and their structure crucially depend on the embedding, making geometric assumptions central to the method.

The authors explicitly state that they do not know how to cast the algorithm in a purely combinatorial setting—i.e., directly in terms of the plane graph’s rotation system and faces without a geometric embedding—thereby posing a concrete unresolved question.