Acyclic Subgraphs of Planar Digraphs

TL;DR

This paper proves a conjecture about the size of acyclic vertex sets in planar digraphs, showing larger acyclic sets exist under certain cycle length conditions, advancing understanding of graph acyclicity.

Contribution

It establishes the conjecture for digraphs with shortest directed cycle length at least 8, generalizing to cycles of length g with a new lower bound.

Findings

Proved the conjecture for digraphs with shortest cycle length ≥ 8

Established a lower bound of (1 - 3/g)n for acyclic sets

Extended results to a broader class of planar digraphs

Abstract

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.