Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs - Tanaka, Hajime (The Electronic Journal of Combinatorics, 2011)
We study $Q$-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width $w$ and 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 $w≥ 2$ 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.