In the computational study of political redistricting, feasibility necessitates the use of a discretization of regions such as states, counties, and towns. In nearly all cases, researchers use a dual graph, whose vertices represent small geographic units (such as census blocks or voting precincts) with edges for geographic adjacency. A political districting plan is a partition of this graph into connected subgraphs that satisfy certain additional properties, such as connectedness, compactness, and equal population. Though dual graphs underlie nearly all computational studies of political redistricting, little is known about their properties. This is a unique graph class that has been described colloquially as `nearly planar, nearly triangulated,' but thus far there has been a lack of evidence to support this description. In this paper we study dual graphs for counties, census tracts, and census block groups across the United States in order to understand and characterize this graph class. We also consider several random graph models (most based on randomly perturbing grids or Delauney triangulations of random point sets), and determine which most closely resemble dual graphs under key metrics. This work lays an initial foundation for understanding and modeling the properties of dual graphs; this will provide invaluable insight to researchers developing algorithms using them to understand, assess, and quantify the properties of political districting plans.

翻译：在政治选区重划的计算研究中，可行性要求对州、县、镇等区域进行离散化处理。几乎所有研究中都采用对偶图，其顶点代表小型地理单元（如人口普查区块或投票选区），边表示地理相邻关系。政治选区划分方案是将该图划分为满足连通性、紧凑性和人口均等性等附加条件的连通子图。尽管对偶图是几乎所有政治选区重划计算研究的基础，但其性质仍知之甚少。这类独特图类常被非正式描述为"近平面、近三角剖分"，但迄今缺乏证据支持该论断。本文研究美国县级、人口普查区级和人口普查区块组级的对偶图，以理解和刻画该图类特征。我们同时考虑多种随机图模型（多数基于随机扰动网格或随机点集的Delaunay三角剖分），并通过关键指标确定最接近真实对偶图的模型。本研究为理解与建模对偶图性质奠定基础，将为研究人员开发用于理解、评估和量化政治选区划分方案性质的算法提供重要洞见。