Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces

Patrizio Angelini, Carla Binucci, Giuseppe Di Battista, Emilio Di Giacomo, Walter Didimo, Fabrizio Grosso, Giacomo Ortali, Ioannis G. Tollis

TL;DR

This paper extends the study of unit-edge-length rectilinear drawings with rectangular faces to non-planar graphs by treating edge crossings as dummy vertices on a grid, enabling grid-based representations where all faces (including the exterior) are rectangles. It introduces two models, UER-RF and UER-USF, and develops efficient, implementable recognition and construction algorithms under various constraints, including preserving rotation systems and prescribed external cycles. Key results include an $O(n)$-time algorithm for testing and constructing UER-USF drawings, polynomial-time algorithms for several restricted UER-RF settings, and an $O(3^k n^{4.5})$-time fixed-parameter tractable algorithm parameterized by the number $k$ of degree-$3$ vertices for the general case. The work suggests that unit-length, grid-based orthogonal drawings with rectangular faces can be efficiently realized even beyond planarity, with practical implications for compact graph visualizations. All findings are framed within a robust theoretical model that unifies geometry, combinatorics, and algorithm design for this drawing paradigm.

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.

Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces

TL;DR

This paper extends the study of unit-edge-length rectilinear drawings with rectangular faces to non-planar graphs by treating edge crossings as dummy vertices on a grid, enabling grid-based representations where all faces (including the exterior) are rectangles. It introduces two models, UER-RF and UER-USF, and develops efficient, implementable recognition and construction algorithms under various constraints, including preserving rotation systems and prescribed external cycles. Key results include an -time algorithm for testing and constructing UER-USF drawings, polynomial-time algorithms for several restricted UER-RF settings, and an -time fixed-parameter tractable algorithm parameterized by the number of degree- vertices for the general case. The work suggests that unit-length, grid-based orthogonal drawings with rectangular faces can be efficiently realized even beyond planarity, with practical implications for compact graph visualizations. All findings are framed within a robust theoretical model that unifies geometry, combinatorics, and algorithm design for this drawing paradigm.

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.

Paper Structure

This paper contains 2 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions

Figures (2)

  • Figure 1: UER-RF drawings: with Unit Square Faces (a), with no internal degree-3 vertex (b), and in the general setting (c). All drawings occupy a $5 \times 5$ grid area.
  • Figure 2: The same graphs depicted in \ref{['fig:crd-examples']} drawn with the topology-shape-metrics algorithm implemented in the GDToolkit library DBLP:reference/crc/BattistaD13. Below each drawing we report the width and the height in terms of grid units. These dimensions are at least twice those of the drawings in \ref{['fig:crd-examples']}; the three drawings have been scaled-down to fit in the same row.