Perfect graphs with polynomially computable kernels

TL;DR

This paper proves new polynomial-time algorithms for computing kernels in certain classes of perfect graphs, including claw-free, chordal, and circular-arc graphs, addressing an open complexity question.

Contribution

It introduces polynomial algorithms for kernel computation in specific perfect graph subclasses, expanding understanding of kernel complexity in directed graphs.

Findings

Polynomial algorithms for claw-free perfect graphs

Kernel computation in chordal graphs is polynomial

Deciding kernel existence in circular-arc graphs is polynomial

Abstract

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.