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{\bf Hajime Tanaka }
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{\bf Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs}
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We study $Q$-polynomial distance-regular graphs from the point of view of what we call \emph{descendents}, that is to say, those vertex subsets with the property that the \emph{width} $w$ and \emph{dual width} $w^*$ satisfy $w+w^*=d$, where $d$ is the diameter of the graph.
We show among other results that a nontrivial descendent with $w\geq 2$ is convex precisely when the graph has classical parameters.
The classification of descendents has been done for the $5$ classical families of graphs associated with short regular semilattices.
We revisit and characterize these families in terms of posets consisting of descendents, and extend the classification to all of the $15$ known infinite families with classical parameters and with unbounded diameter.
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