Simplicial Complex Closure Properties Detector

A pure $d$-dimensional simplicial complex $\Delta$ is closed if there exists a vertex labeling of $\Delta$ such that for every pair of $d$-faces $F=a_0a_1\dots a_d$ and $G=b_0b_1\dots b_d$ (written in increasing order) with $a_i=b_i$ for some $i$, the complex $\Delta$ contains the full $d$-skeleton of the simplex on $F \cup G$.

Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. The complex $\Delta$ is called unit-interval if there exists a labeling $1, \ldots, n$ of its vertices such that for any $d$-face $F=a_0 a_1 \cdots a_d$ (written in increasing order) of $\Delta$, the complex $\Delta$ contains the whole $d$-skeleton of the simplex with vertex set $\{a_0, a_0 + 1, a_0 + 2, \ldots, a_d\}$.

A pure $d$-dimensional simplicial complex $\Delta$ is under-closed if there is a vertex labeling of $\Delta$ such that for every $d$-face $F=a_0a_1\dots a_d$ (written in increasing order) the complex $\Delta$ contains all faces of the form $a_0b_1b_2\dots b_d$ with $b_1\leq a_1$, $b_2\leq a_2$, $\dots$, $b_d\leq a_d$.

A pure $d$-dimensional simplicial complex $\Delta$ is semi-closed if there is a vertex labeling of $\Delta$ such that for every $d$-face $F=a_0a_1\dots a_d$ (written with $a_0 < a_1 < \dots < a_d$) at least one of the following conditions hold:

1. the complex $\Delta$ contains all faces of the form $a_0b_1b_2\dots b_d$ with $b_1\leq a_1$, $b_2\leq a_2$, $\dots$, $b_d\leq a_d$, or
2. the complex $\Delta$ contains all faces of the form $i_0i_1 \dots i_{d-1}a_d$ with $i_0\geq a_0$, $i_1\geq a_1$, $\dots$, $i_{d-1} \geq a_{d-1}$.

A pure $d$-dimensional simplicial complex $\Delta$ is weakly-closed if there is a vertex labeling of $\Delta$ such that for every $d$-face $F=a_0a_1\dots a_d$ (written with $a_0 < a_1 < \dots < a_d$) and for every $g\notin F$ with $a_0 < g < a_d$ there exists a $d$-face $G$ adjacent to $F$ containing $g$ such that either $\max G \neq \max F$ or $\min G \neq \min F$.

A pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is weakly-traceable if there is a vertex labeling of $\Delta$ such that $\Delta$ contains a subset $H_{i_1}$, $H_{i_2}$, $\dots$, $H_{i_k}$ of $\{H_1, H_2 \dots , H_{n-d} \}$ that

1. covers all the vertices and
2. $H_{i_{j}}$ is incident (non-empty intersection) to $H_{i_{j+1}}$ for each $j \in \{1, 2, \dots , k-1\}$.

Here $H_i$ is the facet $(i, i+1, \dots , i+d)$.

A pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is traceable if there is a vertex labeling of $\Delta$ such that $\Delta$ contains facets $H_1$, $H_2$, $\dots$, $H_{n-d}$. Here $H_i$ is the facet $(i, i+1, \dots , i+d)$.

A pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is weakly-Hamiltonian if it has a labeling such that $\Delta$ contains a subset $H_{i_1}$, $\dots$, $H_{i_k}$ of $\{H_1, \dots , H_n\}$ that

1. altogether cover all vertices,
2. $H_{i_j}$ is incident to $H_{i_{j+1}}$ for each $j \in \{1, \dots , k-1\}$,
3. $H_{i_k}$ is incident to $H_{i_1}$.

Here $H_i$ is the facet $(i , i+1 , \dots , i+d)$ using modulo $n$ for anything greater than $n$. For instance, $H_n = (1,2,3, \dots d, n)$.

A pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is Hamiltonian if there is a vertex labeling of $\Delta$ such that $\Delta$ contains facets $H_1$, $H_2$, $\dots$, $H_{n}$. Here $H_i$ is the facet $(i , i+1 , \dots , i+d)$ using modulo $n$ for anything greater than $n$. For instance, $H_n = (1,2,3, \dots d, n)$.

A pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is weakly-chordal if it has a labeling such that, for every two facets $F \ne G$ in $\Delta$ with the same maximum, $\Delta$ also contains some facet $H$ such that $H \subseteq F \cup G - \{ \max F\}$.

A pure $d$-dimensional simplicial complex $\Delta$ is skeleton-weakly-chordal if there is a vertex labeling of $\Delta$ such that the $k$-skeleton of $\Delta$ meets the weakly-chordal condition for all $k\in \{1, 2, \dots, d\}$.

A pure $d$-dimensional simplicial complex $\Delta$ is E-chordal if there is a vertex labeling of $\Delta$ such that for every pair of $d$-faces $F=a_0a_1\dots a_d$ and $G=b_0b_1\dots b_d$ (written in increasing order) with $a_d=b_d$, the complex $\Delta$ contains the whole $d$-skeleton of the simplex on $F \cup G$.