Planar digraphs without large acyclic sets

TL;DR

This paper constructs specific planar directed graphs demonstrating that the largest acyclic set can be at most roughly half the vertices, disproving a previous conjecture and answering an open question.

Contribution

It provides explicit examples of planar digraphs with small maximum acyclic sets, disproving a conjecture and confirming the optimality of a known bound.

Findings

Existence of planar digraphs with maximum acyclic set size at most (n+1)/2

Disproof of Harutyunyan's conjecture

Confirmation that Albertson's question is best possible

Abstract

Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.