  In a directed graph, a kernel is a subset of vertices that is both stable and
absorbing. Not all digraphs have a kernel, but a theorem due to Boros and
Gurvich guarantees the existence of a kernel in every clique-acyclic
orientation of a perfect graph. However, an open question is the complexity
status of the computation of a kernel in such a digraph. Our main contribution
is to prove new polynomiality results for subfamilies of perfect graphs, among
which are claw-free perfect graphs and chordal graphs. Our results are based on
the design of kernel computation methods with respect to two graph operations:
clique-cutset decomposition and augmentation of flat edges. We also prove that
deciding the existence of a kernel - and computing it if it exists - is
polynomial in every orientation of a chordal or a circular-arc graph, even not
clique-acyclic.
