Given a graph G and a parameter r, we define the r-local matroid of G to be the matroid generated by its cycles of length at most r. Extending Whitney’s abstract planar duality theorem from 1932, we prove that for every r the r-local matroid of G is co-graphic if and only if G admits a certain type of embedding in a surface, which we call r-planar embedding. The maximum value of r such that a graph G admits an r-planar embedding is closely related to face-width, and such embeddings for this maximum value of r are quite often embeddings of minimum genus. Unlike minimum genus embeddings, these r-planar embeddings can be computed in polynomial time. This provides the first systematic and polynomially computable method to construct for every graph G a surface so that G embeds in that surface in an optimal way (phrased in our notion of r-planarity).

给定图 G 和参数 r，我们将 G 的 r 局部拟阵定义为由其长度至多为 r 的循环生成的拟阵。扩展 1932 年惠特尼的抽象平面对偶定理，我们证明对于每个 r，G 的 r 局部拟阵是共图的，当且仅当 G 允许某种类型的嵌入在表面中，我们称之为 r 平面嵌入。使得图 G 允许 r 平面嵌入的 r 最大值与面宽密切相关，并且 r 最大值的这种嵌入通常是最小亏格的嵌入。与最小属嵌入不同，这些 r 平面嵌入可以在多项式时间内计算。这提供了第一个系统的、多项式可计算的方法来为每个图 G 构造一个表面，以便 G 以最佳方式嵌入该表面（用我们的 r 平面性概念来表述）。