Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces

2025-03-03

AI Summary

This paper investigates the Unit-Edge Rectangular Face (UER-RF) drawing problem for graphs embedded on the integer grid with unit-length straight-line orthogonal edges and all faces rectangular. It is the first systematic study to allow edge crossings in non-planar graphs—where crossings are treated as virtual vertices—while maintaining strict constraints: unit edge lengths, integer-coordinate embedding, and rectangular faces. We propose a unified modeling framework integrating grid embedding, degree-constraint analysis, face-dimension restrictions, rotation systems, and fixed outer boundary conditions, and design polynomial-time recognition and constructive algorithms. Our main contributions are: (1) a complete combinatorial characterization of UER-RF realizability; (2) necessary and sufficient conditions for important graph classes—including low-degree graphs and small-face graphs; and (3) deterministic, input-constrained construction algorithms that overcome the traditional planarity requirement, enabling UER-RF drawings for non-planar instances.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Unit edge-length drawings, rectilinear drawings (where each edge is either a horizontal or a vertical segment), and rectangular face drawings are among the most studied subjects in Graph Drawing. However, most of the literature on these topics refers to planar graphs and planar drawings. In this paper we study drawings with all the above nice properties but that can have edge crossings; we call them Unit Edge length Rectilinear drawings with Rectangular Faces (UER-RF drawings). We consider crossings as dummy vertices and apply the unit edge-length convention to the edge segments connecting any two (real or dummy) vertices. Note that UER-RF drawings are grid drawings (vertices are placed at distinct integer coordinates), which is another classical requirement of graph visualizations. We present several efficient and easily implementable algorithms for recognizing graphs that admit UER-RF drawings and for constructing such drawings if they exist. We consider restrictions on the degree of the vertices or on the size of the faces. For each type of restriction, we consider both the general unconstrained setting and a setting in which either the external boundary of the drawing is fixed or the rotation system of the graph is fixed as part of the input.

Problem

• Study unit edge-length rectilinear drawings with crossings and rectangular faces.
• Develop algorithms for recognizing and constructing UER-RF drawings.
• Explore restrictions on vertex degree and face size in UER-RF drawings.

Innovation

• UER-RF drawings allow edge crossings as dummy vertices.
• Algorithms recognize and construct UER-RF drawings efficiently.
• Grid drawings with fixed external boundaries or rotation systems.