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Let $G$ be a chordal graph, $I(G)$ be its independence complex, and $v \in V(G)$ be a simplicial vertex (that is, $v$'s neighborhood is a clique).

Is there a way to relate the homology of $I(G)$ and the homology of $I(G \setminus v)$? Ideally, I'd like to be able to compute the homology of $I(G \setminus v)$ from the homology of $I(G)$.

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  • $\begingroup$ There should be a Mayer-Vietoris long exact sequence relating the homology of $I(G), I(G\setminus V)$, and $I(G\setminus N(v))$ where $N(v)$ is the neighborhood of $v$. None of your assumptions seem to simplify the calculation of $I(G\setminus N(v))$ so this may not be very useful for what you want. $\endgroup$
    – Will Sawin
    Commented Mar 21 at 0:40

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There is a fairly standard splitting technique as suggested in the comment, although not necessarily using $v$.

Let $G$ be a graph and $v$ is a simplicial vertex. Let $w$ be any neighbor of $v$. Then you have a homotopy equivalence $$I(G)\simeq I(G\setminus w) \vee \Sigma I(G\setminus N_G[w])$$ where $N_G[w]$ denotes the closed neighborhood of $w$ (including $w$ itself).

The proof uses the argument that the inclusion of the link of $w$ in the deletion of $w$ in $I(G)$ is null-homotopic since it factors through a cone with apex $v$.

See for example Theorem 3.7 in https://arxiv.org/abs/math/0508148 (Alex Engstrom) for this specific case or section 3 of https://arxiv.org/abs/1106.6250 (myself) for a slightly broader discussion of this technique with some more general conditions on $v$. You may also be interested in the paper https://core.ac.uk/download/pdf/82040661.pdf (Kazuhiro Kawamura).

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  • $\begingroup$ Thanks. With this answer I was able to turn the Mayer-Vietoris sequence into something useful for my problem. $\endgroup$
    – Will
    Commented Mar 27 at 23:02

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