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arXiv · 2509.04168

Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid

Henry Förster·Stephen Kobourov·Jacob Miller·Johannes Zink
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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.

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Henry Förster, Stephen Kobourov, Jacob Miller, Johannes Zink. 2025-09-04. Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid. https://arxiv.org/abs/2509.04168

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