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The $r$-stable \\emph{interlacing graph} $\\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\\{1,...,n\\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \\emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. 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