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Greedily Constructing Small Quasi-Kernels

Alexander Clow

TL;DR

The paper tackles the problem of constructing small quasi-kernels in digraphs by introducing a polynomial-time greedy algorithm that yields quasi-kernels in sourceless digraphs. It formalizes a general framework for hereditary classes, proving an $O(n^4)$-time algorithm that outputs a quasi-kernel of size at most $\frac{t}{t+1}n$, and derives concrete bounds for several graph families, including $\vec{K}_{1,d}$-free graphs and digraphs with bounded out-degree or restricted short cycles (e.g., girth at least seven), with improved bounds such as $|Q|\le \frac{(d^2-2d+2)n}{d^2-d+1}$ and $|Q|\le \frac{4n}{7}$ for specific cases. The work additionally provides $O(n^2)$-time algorithms under certain cycle-restriction conditions and discusses the limitations of the approach in general, contributing to the understanding of when small quasi-kernels exist and how to efficiently compute them. Overall, the results advance toward the Small Quasi-Kernel Conjecture by delivering constructive methods and concrete bounds for several natural digraph classes, with potential implications for related dominance and independence problems in directed graphs.

Abstract

In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chvátal and Lovász showed every digraph contains a quasi-kernel. In 1976, P. L. Erdős and Székely conjectured that every sourceless digraph has a quasi-kernel of order at most $\frac{n}{2}$. Despite significant recent attention by the community the problem remains far from solved, with no bound of the form $(1-ε)n$ known. We introduce a polynomial time algorithm which greedily constructs a small quasi-kernel. Using this algorithm we show that if $D$ is a $\vec{K}_{1,d}$-free digraph, then $D$ has a quasi-kernel of order at most $\frac{(d^2 - 2d + 2)n}{d^2-d+1}$. By refining this argument we prove that for any $D$ with maximum out-degree $3$ this algorithm constructs a quasi-kernel of order at most ${4n}/{7}$. Finally, we consider the problem in digraphs forbidding certain orientation of short cycles as subgraphs, concluding that all orientations $D$ of a graph $G$ with girth at least $7$ have a quasi-kernel of order at most $\frac{(d^2+4)n}{(d+2)^2}$, where $d$ is the maximum out-degree of $D$.

Greedily Constructing Small Quasi-Kernels

TL;DR

The paper tackles the problem of constructing small quasi-kernels in digraphs by introducing a polynomial-time greedy algorithm that yields quasi-kernels in sourceless digraphs. It formalizes a general framework for hereditary classes, proving an -time algorithm that outputs a quasi-kernel of size at most , and derives concrete bounds for several graph families, including -free graphs and digraphs with bounded out-degree or restricted short cycles (e.g., girth at least seven), with improved bounds such as and for specific cases. The work additionally provides -time algorithms under certain cycle-restriction conditions and discusses the limitations of the approach in general, contributing to the understanding of when small quasi-kernels exist and how to efficiently compute them. Overall, the results advance toward the Small Quasi-Kernel Conjecture by delivering constructive methods and concrete bounds for several natural digraph classes, with potential implications for related dominance and independence problems in directed graphs.

Abstract

In a digraph ,a quasi-kernel is an independent set such that for every vertex , there is a vertex satisfying . In 1974 Chvátal and Lovász showed every digraph contains a quasi-kernel. In 1976, P. L. Erdős and Székely conjectured that every sourceless digraph has a quasi-kernel of order at most . Despite significant recent attention by the community the problem remains far from solved, with no bound of the form known. We introduce a polynomial time algorithm which greedily constructs a small quasi-kernel. Using this algorithm we show that if is a -free digraph, then has a quasi-kernel of order at most . By refining this argument we prove that for any with maximum out-degree this algorithm constructs a quasi-kernel of order at most . Finally, we consider the problem in digraphs forbidding certain orientation of short cycles as subgraphs, concluding that all orientations of a graph with girth at least have a quasi-kernel of order at most , where is the maximum out-degree of .
Paper Structure (7 sections, 13 theorems, 51 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 7 sections, 13 theorems, 51 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Constructing Small Quasi-Kernels
  5. Digraphs with Small Out-Degree
  6. Restricting Short Cycles
  7. Future Work

Key Result

Theorem 1.2

Let $t\geq 1$ be a constant, and let $\mathcal{H}$ be a hereditary class of digraphs such that for all sourceless digraphs $D \in \mathcal{H}$ there exists a vertex $v \in V(D)$ where $v \notin N^+(S(v))$ and or $v \in N^+(S(v))$ and Then, there is a $O(n^4)$ time algorithm which takes any sourceless digraph $D \in \mathcal{H}$ of order $n$ and returns a quasi-kernel $Q$ in $D$ such that $|Q| \l

Figures (3)

  • Figure 1: Some orientations of $C_3,C_4,C_6$, and $K_{1,t}$ that are of interest.
  • Figure 2: The $7$ vertex Paley Tournament with $3$ degree $1$ sinks appended to each vertex.
  • Figure 3: Two figures which display the key cases covered to prove Theorem \ref{['Thm: Out-degree 3']}.

Theorems & Definitions (29)

  • Conjecture 1.1: The Small Quasi-Kernel Conjecture
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['Thm: Main CounterStructure']}
  • Corollary 2.2
  • ...and 19 more