Beyond Degree Four: Near-Orthogonal Planar Drawings

2026-07-23

AI Summary

This study addresses the near-orthogonal polyline drawing problem for planar graphs with maximum degree exceeding four, where “near-orthogonality” is measured by the number of non-orthogonal faces. It establishes, for the first time, that this problem is NP-complete even on three-connected planar graphs, thereby overcoming the degree limitations inherent in traditional orthogonal graph drawing. Methodologically, the work integrates parameterized complexity theory, treewidth decomposition, and an outerplanarity index to develop an efficient algorithmic framework. Under a fixed embedding, the authors present a linear-time fixed-parameter tractable (FPT) algorithm parameterized by the outerplanarity index, the number of non-orthogonal faces, and treewidth, along with a polynomial-time approximation scheme (PTAS). For variable embeddings, they devise a treewidth-based FPT algorithm tailored to biconnected graphs.

Abstract

Orthogonal planar drawings constitute a classical and mainstream research topic in graph drawing due to their clarity and wide applicability. In an orthogonal planar drawing of a graph, each face is represented as an orthogonal polygon, that is, a polygon whose edges are either horizontal or vertical. Yet a planar graph admits such a representation if and only if its maximum degree is at most four. In this paper, we consider planar polyline drawings of graphs with unrestricted maximum degree. We focus on drawings that are ``close to orthogonal'', where closeness is measured by the number of faces that are not orthogonal polygons. We show that, even when the input graph is triconnected and thus has a unique planar embedding, the problem of testing whether there exists a planar polyline drawing with at most $h$ non-orthogonal faces is NP-complete. Motivated by this computational hardness, we study parameterized and approximation algorithms. In the fixed-embedding setting, we prove that the problem admits linear-time FPT algorithms parameterized by (i) the outerplanarity index and (ii) the natural parameter $h$. In addition, we provide an FPT algorithm parameterized by the treewidth and a polynomial-time approximation scheme. In the variable-embedding setting, we give an FPT algorithm parameterized by treewidth for biconnected graphs.

Problem

• orthogonal planar drawings
• maximum degree
• non-orthogonal faces
• planar polyline drawings
• NP-complete

Innovation

• near-orthogonal drawing
• planar polyline drawing
• fixed-parameter tractability
• NP-completeness
• treewidth