Shellability of the lattice-distance subcomplexes

Determine whether each simplicial complex Gamma_s is shellable and whether Gamma admits a shelling order in which every facet of Gamma_s precedes every facet of Gamma_{s+1} for each s=1,...,k-1.

Background

The complexes Gamma_s are generated by facets of a triangulation Gamma whose lattice distance from the deleted-edge vertex is at least a specified threshold. Shellability would provide a combinatorial interpretation of the h-vectors of these subcomplexes.

Such an interpretation could clarify the coefficient structure of the differences h_{Gamma_s}(t)-h_{Gamma_{s-1}}(t), which appear in the conjectured decomposition of the h*-polynomial difference. The authors state the issue as an explicit question motivated by positive computations for graphs with few vertices.

References

Are the simplicial complexes $\Gamma_s$ shellable for any $s= 1, \ldots, k$? Is there a shelling order of $\Gamma$ for which every facet of $\Gamma_s$ is smaller than any facet of $\Gamma_{s+1}$, for every $s=1,\dots,k-1$?

— The number of edges of a symmetric edge polytope  (2512.16572 - Codenotti et al., 18 Dec 2025) in Question, Section 4 (Further directions)