On planarity of graphs in homotopy type theory

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Jonathan Prieto-Cubides, Håkon Robbestad Gylterud
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引用次数: 0

Abstract

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.
论同调类型理论中图形的平面性
在本文中,我们介绍了同调类型理论(HoTT)中图论的构造性和证明相关性发展,包括图的概念、图的面和嵌入球面的图的映射。这使我们能够为局部有向有限多图和连通多图的平面性提供一个基本特征,这个特征从拓扑图理论,特别是从组合图嵌入曲面中得到启发。如果一个图有一个映射和一个外表面,嵌入图中的任何行走都与另一个行走同构,那么这个图就是平面图。结果是,这类平面映射构成了图的同构集。作为归纳构建平面图实例的一种方法,我们引入了平面映射的扩展。我们在支持 HoTT 的证明助手 Agda 中形式化了这项工作的重要部分。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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