The bisimplicial-vertex conjecture for minimally non-perfectly divisible graphs

A graph is minimally non-perfectly divisible (MNPD) if it is not perfectly divisible but every proper induced subgraph is perfectly divisible. A vertex is bisimplicial if its neighbourhood can be covered by two cliques.

Bisimplicial-vertex conjecture. No MNPD graph contains a bisimplicial vertex.

This is proposed using the result that every even-hole-free graph contains a bisimplicial vertex. Its status is open.