Regularity of extended double covers of strongly perfect graphs

Determine whether, for every strongly perfect graph G, the Castelnuovo–Mumford regularity of the extended double cover B_e(G) equals the independence number of G.

Background

The paper proves that reg(B_e(G))=α(G) for several subclasses of strongly perfect graphs, including chordal, co-chordal, cograph, comparability, and bipartite graphs. It then gives a strongly perfect graph for which cochord(B_e(G))>α(G), showing that the broader equality involving the co-chordal covering number fails in this class.

The authors note that the counterexample still satisfies reg(B_e(G))=α(G), leaving open whether this regularity equality holds for all strongly perfect graphs.

References

Is it true that for any strongly perfect graph $G$, the equality $reg(B_e(G))=\alpha(G)$ holds?

— Projective dimension of closed neighborhood hypergraphs via extended double covers  (2610.06556 - Civan et al., 5 Oct 2026) in Section Further Comments, immediately after Example (example:cd-alpha)