How Many Slopes Does Polynomial Area Cost?

2026-05-29

AI Summary

This study addresses the challenge of simultaneously achieving few slopes and few bends per edge in drawings of high-maximum-degree planar graphs while maintaining polynomial area. It systematically uncovers, for the first time, the trade-offs among the number of slopes, the number of bends per edge, and drawing area, thereby filling a critical gap in the literature on polynomial-area, low-slope graph drawings. Leveraging techniques from graph theory and geometric embedding, the authors propose a novel drawing algorithm that significantly reduces edge bends with only a modest increase in the number of slopes, all while guaranteeing polynomial area. The resulting construction outperforms existing approaches that rely on super-polynomial area, offering both theoretical insights and practical tools for the efficient visualization of high-degree planar graphs.

Abstract

In this work, we study the interplay between the number of slopes, the number of bends per edge, and the area requirements for planar drawings of bounded-degree graphs. Our motivation stems from the fact that, while numerous algorithms produce planar drawings with few slopes for graphs of relatively small degree in polynomial area, existing approaches for higher-degree graphs often require super-polynomial area. We address this gap in the literature by presenting new constructions that yield polynomial-area drawings with few bends per edge while slightly increasing the required number of slopes, thereby providing the first systematic study of slopes, bends and area trade-offs.

Problem

• slopes
• bends
• area
• planar drawings
• bounded-degree graphs

Innovation

• slopes
• bends
• polynomial area
• planar drawings
• bounded-degree graphs