Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

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 $(1,0)$, 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 conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.