Counting Kernels in Directed Graphs with Arbitrary Orientations

TL;DR

This paper develops a polynomial-time algorithm for counting kernels in fuzzy circular interval graphs with arbitrary orientations, and explores kernel problems in cographs and threshold graphs, expanding understanding of kernel complexity.

Contribution

It introduces a polynomial-time counting algorithm for kernels in fuzzy circular interval graphs with arbitrary orientations, and analyzes kernel problems in cographs and threshold graphs.

Findings

Counting kernels in fuzzy circular interval graphs is polynomial-time solvable.

Kernel counting is NP-hard in general cographs.

Threshold graphs allow linear-time kernel counting.

Abstract

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel). Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem all restrict both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005). By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.) We also consider kernels on cographs, where we establish NP-hardness in general but linear running times on the subclass of threshold graphs.