Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid
Computational Geometry
2025-09-05 v1 Discrete Mathematics
Abstract
A strengthened version of Harborth's well-known conjecture -- known as Kleber's conjecture -- states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the integer grid. Positive results for Kleber's conjecture are known for planar 3-regular graphs, for planar graphs that have maximum degree 4, and for planar 3-trees. However, all but one of the existing results are existential and do not provide bounds on the required grid size. In this paper, we provide polynomial-time algorithms for computing crossing-free straight-line drawings of trees and cactus graphs with integer edge lengths and integer vertex position on polynomial-size integer grids.
Cite
@article{arxiv.2509.04168,
title = {Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid},
author = {Henry Förster and Stephen Kobourov and Jacob Miller and Johannes Zink},
journal= {arXiv preprint arXiv:2509.04168},
year = {2025}
}