A minimally nonperfectly divisible graph with a bisimplicial vertex

We disprove Ho\`ang's conjecture that a minimally nonperfectly divisible graph cannot contain a bisimplicial vertex. Our counterexample has 93 vertices, 320 edges, clique number three, and a bisimplicial vertex of degree four. A perfect division of a graph $H$ is a partition $V(H)=A\mathbin{\dot\cup}B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$; a graph is perfectly divisible when each of its induced subgraphs with at least one edge has such a division. The constructed graph has no perfect division, whereas every proper induced subgraph is perfectly divisible. Its construction uses a forcing mechanism based on rooted graphs: six 15-vertex rooted graphs force their roots into $A$, and a nine-vertex auxiliary graph then forces a triangle into $B$. We isolate this mechanism in a rooted composition lemma, which proves symbolically that every proper induced subgraph is perfectly divisible. All finite assertions concerning the rooted and auxiliary graphs are verified by exact exhaustive computation, with an independent implementation providing a cross-check. The same construction gives a negative answer to a prescribed-vertex problem of Hu, Xu and Zhuang.