$\DeclareMathOperator\width{width}$Let $P$ be a finite poset with $n$ elements (we can assume that $P$ is connected and has width at most $n-2$). The **comparability graph** $G_P=(V,E)$ associated to $P$ is by definition the finite graph with vertices $V=P$ and two elements $v, w \in V$ are adjacent if $v<w$ or $w<v$.
A subset $S$ of $V$ is called a **dominating set in $G$** if every vertex $v$ in $G$ belongs to $S$ or is adjacent to an element of $S$.

The **dominance complex $D(G_P)$** is the simplicial complex consisting of the subsets of $V$ whose complements are dominating. We look at homology of such a simplicial complex over fields here (see Perkinson - Homology of Simplicial Complexes for a definition).

Question (Formulation 1): Is it true that the first degree in which the reduced homology is non-zero of $D(G_P)$ is equal to $n-1-\width(P)$?

(Here $\width(P)$ denotes the maximal cardinality of an antichain in the poset $P$.)

By theorem 1 in Matsushita - Dominance complex and vertex cover number, we should have that the first non-zero degree of the reduced homology is less than or equal to $n-1-\width(P)$. And the question asks whether we have equality in theorem 1 of the previous article in the case of graphs given as comparability graphs of finite posets.

Here is a more direct alternative formulation of the problem:

Let $P$ be a finite poset with $n$ elements (we can assume that $P$ is connected and the width of $P$ is at most $n-2$). For $p \in P$ set $J(p):=\{q \in P \mid p \nleq q \}$ and $I(p):= \{ q \in P \mid q \leq p \}$. For a subset $S \subseteq P$ set $J(S) := \bigcap\limits_{p \in S}{J(p)}$ and $I(S):= \bigcup\limits_{p \in S}{I(p)}$.

Then the simplicial complex $\Gamma(P)$ associated to $P$ is defined by the condition $S \in \Gamma(P)$ if and only if $J(S^c) \subseteq I(S^c)$.

Question (Formulation 2): Is it true that the first non-zero positive degree of the homology of $\Gamma(P)$ appears at $n-1-\width(P)$?

The question is tested with a computer and true for all posets with at most 10 elements (thus for nearly 3 million examples).

(Background: The original formulation of the question is Formulation 2 and the connection to the article Matsushita - Dominance complex and vertex cover number was noted by Hugh Thomas.)

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