发表机构
KU Leuven; Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences; Faculty of Information Technology, Czech Technical University in Prague; School of Mathematical Sciences, Lancaster University, UK（荷语鲁汶大学; 奥地利科学院约翰·拉德纳计算与应用数学研究所; 布拉格捷克理工大学信息技术学院; 兰开斯特大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单位距离图及其子类（便士图、弹珠图、火柴杆图）的刚性性质，回答了一个关于便士图和弹珠图刚性的开放问题，并建立了便士图与NAC着色之间的联系。
AI 中文摘要
单位距离图是一种在欧几里得空间中可实现的图，其中每条边的长度均为单位长度。对非边施加进一步的几何条件，可以得到一系列自然的子类。要求任意两个顶点之间的距离不小于1，可得到便士图和弹珠图：即$d=2,3$维空间中具有不相交内部的等半径$d$维球体的接触图。相反，要求平面中的直线绘制为非交叉，则得到火柴杆图。由于实现的任何运动都必须保持这些额外条件，所得框架的刚性和柔性性质不同于经典的杆-关节刚性理论。本文分析了便士图、弹珠图、火柴杆图和单位距离图的多种刚性相关问题。特别地，我们回答了关于便士图和弹珠图刚性的一个近期开放问题，并建立了便士图与NAC着色概念之间的联系。
英文摘要
A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a family of natural subclasses. Requiring that no two vertices lie at distance less than one gives penny and marble graphs: the contact graphs of collections of equal radii $d$-dimensional spheres with non-overlapping interiors for $d=2,3$. Requiring instead that the straight-line drawing in the plane be non-crossing gives matchstick graphs. Since any motion of a realisation must preserve these extra conditions, the rigidity and flexibility properties of the resulting frameworks differ from those of classical bar-joint rigidity theory. In this note we analyse various rigidity problems for penny and marble graphs, matchstick graphs and unit-distance graphs. In particular we answer a recent open problem on penny and marble graph rigidity and establish a link between penny graphs and the concept of NAC-colourings.
