Kernel-perfect directed graph

A digraph kernel is an independent set $K$ of vertices such that each vertex outside $K$ has a directed edge from itself to a vertex in $K$. A directed graph is kernel-perfect if every induced subgraph has a digraph kernel. Preference orientations of the line graph of a bipartite graph are kernel-perfect: a stable matching in a bipartite graph is independent in the line graph, and stability supplies the required outgoing edge at every unmatched edge.