Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs
Therese Biedl, Veronika Irvine

TL;DR
This paper explores the topological and combinatorial properties of graph drawings inspired by bobbin lace patterns, providing algorithms for verifying embeddings and constructing straight-line drawings on a torus.
Contribution
It introduces a linear-time verification method for specific graph embeddings and an algorithm to find fundamental cycles for creating lace pattern drawings on a torus.
Findings
Verification of combinatorial embeddings can be done in linear time.
A method to find fundamental cycles in a supergraph is provided.
A straight-line drawing within a rectangular frame is achievable for lace graphs.
Abstract
In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to , that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Geometric and Algebraic Topology · Topological and Geometric Data Analysis
