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Counting Kernels in Directed Graphs with Arbitrary Orientations

Bruno Jartoux

TL;DR

The paper addresses the problem of counting kernels in arbitrarily oriented digraphs, focusing on fuzzy circular interval graphs (FCIGs) to achieve tractability where general cases are hard. It introduces a dynamic-programming framework for fuzzy linear interval graphs (FLIGs) based on a weak vertex ordering, and then extends to FCIGs by reducing circular cases to linear subproblems via a recursive kernel description encoded as a labelled acyclic digraph. The results yield polynomial-time algorithms for counting kernels (and producing one when it exists) and have corollaries for counting maximal and maximum independent sets in FCIGs; extensions to weights and constrained subsets are also developed. The work illuminates a tractable frontier for kernel-related problems in claw-free graphs with arbitrary orientations and connects to broader themes such as MIS counting and graph-width parameters, with implications for several related graph classes (e.g., cographs, threshold graphs, concave-round graphs).

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.

Counting Kernels in Directed Graphs with Arbitrary Orientations

TL;DR

The paper addresses the problem of counting kernels in arbitrarily oriented digraphs, focusing on fuzzy circular interval graphs (FCIGs) to achieve tractability where general cases are hard. It introduces a dynamic-programming framework for fuzzy linear interval graphs (FLIGs) based on a weak vertex ordering, and then extends to FCIGs by reducing circular cases to linear subproblems via a recursive kernel description encoded as a labelled acyclic digraph. The results yield polynomial-time algorithms for counting kernels (and producing one when it exists) and have corollaries for counting maximal and maximum independent sets in FCIGs; extensions to weights and constrained subsets are also developed. The work illuminates a tractable frontier for kernel-related problems in claw-free graphs with arbitrary orientations and connects to broader themes such as MIS counting and graph-width parameters, with implications for several related graph classes (e.g., cographs, threshold graphs, concave-round graphs).

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.
Paper Structure (28 sections, 17 theorems, 17 equations, 3 figures, 1 algorithm)

This paper contains 28 sections, 17 theorems, 17 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background.
  3. Kernels in specific digraph classes.
  4. Notation and terminology.
  5. Counting kernels in fuzzy circular interval graphs
  6. Fuzzy circular interval graphs and subclasses
  7. Nice models.
  8. Roadmap.
  9. The structure of kernels in fuzzy linear interval graphs
  10. Weak vertex ordering.
  11. Preliminary facts.
  12. Recursive description of the kernels.
  13. The recursive description as a labelled acyclic digraph.
  14. Reducing the circular case to the linear case
  15. Inapplicability of our approach to general interval graphs.
  16. ...and 13 more sections

Key Result

Theorem 1

Given an $n$-vertex fuzzy circular interval graph $D$ with arbitrary edge orientations, we can compute the size of $\ker D$ and, if nonempty, return one of its elements in time polynomialSee runtimes for a discussion of running times. They are $\mathcal{O}(\lvert D\rvert^3\cdot \lvert\lvert D \rvert

Figures (3)

  • Figure 1: Proper inclusions among some graph classes. Inclusions without a reference are well-known or obvious. \ref{['fig:FLIGexample']} shows a fuzzy linear interval graph which is not circular arc, and the claw $K_{1,3}$ itself is an interval graph that is not claw-free. The Information System on Graph Classes and their Inclusions is a helpful resource InformationSystemGraph.
  • Figure 2: A graph and a nice FLIG model. The reader can check that this graph is not an LIG, nor more generally a circular arc graph. Hint: the disjoint union of a 4-cycle and a vertex is not a circular arc graph.
  • Figure 3: Left: the FCIG $D$ with a nice model. The vertices $a$ and $b$ are not adjacent and absorb $[a;b]$. Right: the subgraph $D'$ obtained by deleting $\operatorname{N}^{-}(\{a,b\})$ has a nice FLIG model in which $a$ and $b$ are extremal, deduced from the original model by shrinking or deleting intervals without perturbing adjacencies.

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 21 more