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

2025-09-04

AI Summary

This paper addresses the problem of crossing-free straight-line embeddings of planar graphs on integer grids, specifically focusing on trees and cactus graphs with **integer edge lengths**. To resolve the lack of constructive algorithms for Kleber’s conjecture, we propose the first polynomial-time algorithm. Our method integrates computational geometry and graph theory, employing a recursive configuration scheme coupled with coordinate scaling to guarantee that all vertices lie on integer lattice points, all edges have integer lengths, and no edges cross. The embedding is realized within an $O(n^c)$-sized integer grid (where $c$ is a constant), markedly improving upon prior exponential-grid upper bounds. This constitutes the first nontrivial graph family—beyond trivial cases—for which crossing-free integer-length embeddings are constructible in polynomial-size integer grids. The result provides a foundational constructive tool for integer geometric graph embedding theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Drawing trees with integer edge lengths
• Drawing cactus graphs on integer grids
• Polynomial-time algorithms for grid drawings

Innovation

• Polynomial-time algorithms for integer edge drawings
• Polynomial-size integer grid vertex positioning
• Crossing-free straight-line drawings for trees and cacti