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Characterize 3-convex graph–nonedge-set pairs using the 3-SIP characterization

Characterize all pairs (G,F), where G is a finite graph and F is a set of nonedges, such that for every squared edge-length map ℓ the Cayley configuration space Ω^3_F(G,ℓ) is convex (i.e., (G,F) is 3-convex), by leveraging the 3-SIP characterization of graph–nonedge pairs in Theorem 3-sip_characterization, analogous to the known 2D characterization of 2-convex graph–nonedge-set pairs.

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Background

For d ≤ 2, prior work provides characterizations of both d-SIP for pairs (G,f) and d-convexity for sets (G,F). This paper extends the SIP side to d = 3 via an f-preserving forbidden-minor condition localized to atoms. A comparable 3D characterization for 3-convex pairs (G,F) remains to be developed.

The open problem asks to use the new 3-SIP characterization as a foundation to obtain a full characterization of 3-convex pairs (G,F), mirroring how the 2D convexity characterization was built on the 2-SIP characterization.

References

Open Problem. Use the characterization in Theorem \ref{thm:3-sip_characterization} to characterize $3$-convex graph-nonege-set pairs, similar to how the characterization of $2$-convex graph-nonege-set pairs in is based on their characterization of graph-nonege pairs with the $2$-SIP.

— Graphs with single interval Cayley configuration spaces in 3-dimensions (2409.14227 - Sims et al., 21 Sep 2024) in Section 6 (Open problems and conjectures)