Develop efficient recognition and complexity results for unswappability

Determine whether unswappability of a graph can be checked more efficiently than by testing graph isomorphism after every admissible edge swap, and establish a computational hardness result if such an efficient procedure does not exist.

Background

Unswappability is central to the paper’s lower-bound method: if swapping two disjoint same-labeled edges produces a nonisomorphic graph, then those edges cannot share a bond-edge type in a valid Scenario 3 construction. The paper notes that checking this property may require repeated graph-isomorphism comparisons and that its appendix checker uses spanning-tree counts only as a heuristic. It therefore leaves open both the algorithmic design question and the associated complexity-theoretic question.