Polynomial-time decision procedure for the 3-SIP characterization

Develop a polynomial-time algorithm that, given a finite graph G and a nonedge f, decides whether no atom of G ∪ f that contains f has an f-preserving 3-forbidden minor (i.e., a K5 or K2,2,2 minor preserving f), equivalently deciding whether the graph–nonedge pair (G,f) has the 3-single interval property (3-SIP) characterized in Theorem 3-sip_characterization.

Background

The paper proves a forbidden-minor-based characterization (Theorem 3-sip_characterization) of when a graph–nonedge pair (G,f) has the 3-single interval property (3-SIP): namely, iff no atom of G ∪ f that contains f has an f-preserving 3-forbidden minor. The 3-forbidden minors are K5 and K2,2,2. Atoms are the vertex-maximal induced subgraphs with no clique separators.

While the characterization is structural, an efficient decision procedure is not provided. The authors suggest that techniques inspired by the polynomial-time algorithms for fixed finite excluded-minor families (e.g., Robertson–Seymour theory) might be adapted, but the f-preserving condition and atom-localization make the algorithmic problem nontrivial.