2026/07/23 by Bruno Benedetti, Marta Pavelka
Mathematics · #math.CO
We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) Δ skeleton-E-chordal ⇒ Δ^\vee vertex-decomposable ⇒ Δ skeleton-clique-chordal. Moreover, for subflag complexes, Δ skeleton-E-chordal \Longleftrightarrow Δ^\vee vertex-decomposable. (For d=1 this boils down to ``G chordal \Longleftrightarrow G^\vee vertex-decomposable'', a result closely related to Fröberg's theorem.) (2) For subflag complexes, Δ is skeleton-E-chordal \Longleftrightarrow it splits as Δ= Δ1 ∪ Δ2, with each Δi a skeleton-E-chordal induced subcomplex of Δ, and with Δ1 ∩ Δ2 a complex whose 1-skeleton is a clique. (This generalizes ``G chordal \Longleftrightarrow G splits as a union of chordal graphs that intersect in a common clique''). (3) Δ skeleton-E-chordal \Longleftrightarrow every nonempty induced subcomplex of Δ has a skeleton-E-simplicial vertex. (Generalizes ``G chordal ⇔ every nonempty induced subgraph has a simplicial vertex''.) (4) Δ underclosed ⇒ Δ skeleton-weakly-chordal and weakly-closed. (Generalizes ``G interval ⇒ G chordal and co-comparability''.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)