Sandwich monotonicity of weakly chordal graphs

Determine whether the class of weakly chordal graphs is sandwich monotone; specifically, decide if for every weakly chordal graph G and every non-empty set F of edges such that G − F remains weakly chordal, there must exist at least one edge e in F for which G − e is also weakly chordal.

Background

The paper studies recognition of weighted graph classes via level graphs and edge elimination schemes, introducing and leveraging the notion of degree sandwich monotone classes. Several classes, including split, threshold, and chain graphs, are shown to be degree sandwich monotone, which enables linear-time recognition of their weighted analogs.

Sandwich monotonicity was previously established for classes such as chordal, strongly chordal, and chordal bipartite graphs. Whether weakly chordal graphs enjoy sandwich monotonicity remains unresolved; resolving this would clarify the structural behavior of weakly chordal graphs under edge deletions and potentially impact algorithmic recognition strategies for associated weighted versions.