Deciding monotonicity of simple drawings of the complete graph

Oswin Aichholzer; Thomas Hackl; Alexander Pilz; Gelasio Salazar; Birgit Vogtenhuber

Deciding monotonicity of simple drawings of the complete graph

Computational Geometry 2026-07-07 v1 Combinatorics

Authors: Oswin Aichholzer , Thomas Hackl , Alexander Pilz , Gelasio Salazar , Birgit Vogtenhuber

Abstract

A drawing of a graph is {\em $x$ -monotone} if every vertical line intersects each edge of the graph at most once. We present an $O(n^5)$ time algorithm for deciding whether a simple drawing of the complete graph $K_n$ is weakly isomorphic to an $x$ -monotone drawing. We note that this algorithm can also decide whether a drawing of $K_n$ is strongly isomorphic to an $x$ -monotone drawing.

Keywords

graph drawing x-monotone complete graph isomorphism algorithm

Cite

@article{arxiv.2607.06240,
  title  = {Deciding monotonicity of simple drawings of the complete graph},
  author = {Oswin Aichholzer and Thomas Hackl and Alexander Pilz and Gelasio Salazar and Birgit Vogtenhuber},
  journal= {arXiv preprint arXiv:2607.06240},
  year   = {2026}
}