Skeleton Chordalities

Abstract

We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) $\hat{I}”$ skeleton-E-chordal $\Rightarrow$ $\hat{I}{”}^{\vee }$ vertex-decomposable $\Rightarrow$ $\hat{I}”$ skeleton-clique-chordal. Moreover, for subflag complexes, $\hat{I}”$ skeleton-E-chordal $⟺$ $\hat{I}{”}^{\vee }$ vertex-decomposable. (For $d=1$ this boils down to ``$G$ chordal $⟺$ $G^{\vee }$ vertex-decomposable'', a result closely related to Fröberg's theorem.) (2) For subflag complexes, $\hat{I}”$ is skeleton-E-chordal $⟺$ it splits as $\hat{I}”=\hat{I}{”}_1\cup \hat{I}{”}_2$, with each $\hat{I}{”}_i$ a skeleton-E-chordal induced subcomplex of $\hat{I}”$, and with $\hat{I}{”}_1\cap \hat{I}{”}_2$ a complex whose $1$-skeleton is a clique. (This generalizes ``$G$ chordal $⟺$ $G$ splits as a union of chordal graphs that intersect in a common clique''). (3) $\hat{I}”$ skeleton-E-chordal $⟺$ every nonempty induced subcomplex of $\hat{I}”$ has a skeleton-E-simplicial vertex. (Generalizes ``$G$ chordal $\iff$ every nonempty induced subgraph has a simplicial vertex''.) (4) $\hat{I}”$ underclosed $\Rightarrow$ $\hat{I}”$ skeleton-weakly-chordal and weakly-closed. (Generalizes ``$G$ interval $\Rightarrow$ $G$ 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.)