Probabilistic Matching of Planar Regions

关于二层k-匹配平面图

On 2-Layer k-Matching-Planar Graphs

研究二层k-匹配平面图，证明此类图路径宽度至多为2k + 1并构造达到3⌊k/2⌋ + 1的图，探讨单边识别问题的NP难及固定参数可处理性，还证明双边变体在多项式时间内不能常数因子近似。

Large Matchings in Maximal 1-planar graphs

It is well-known that every maximal planar graph has a matching of size at least $\tfrac{n+8}{3}$ if $n\geq 14$. In this paper, we investigate similar matching-bounds for maximal \emph{1-planar} graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. In particular we show that every 3-connected simple-maximal 1-planar graph has a matching of size at least $\tfrac{2n+6}{5}$; the bound decreases to $\tfrac{3n+14}{10}$ if the graph need not be 3-connected. We also give (weaker) bounds when the graph comes with a fixed 1-planar drawing or is not simple. All our bounds…

Finding large matchings in 1-planar graphs of minimum degree 3

A matching is a set of edges without common endpoint. It was recently shown that every 1-planar graph (i.e., a graph that can be drawn in the plane with at most one crossing per edge) that has minimum degree 3 has a matching of size at least $\frac{n+12}{7}$, and this is tight for some graphs. The proof did not come with an algorithm to find the matching more efficiently than a general-purpose maximum-matching algorithm. In this paper, we give such an algorithm. More generally, we show that any matching that has no augmenting paths of length 9 or less has size at least $\frac{n+12}{7}$ in a 1-…

平面点集上整数坐标下的多对多匹配的精确亚二次算法

Exact Subquadratic Algorithm for Many-to-Many Matching on Planar Point Sets with Integer Coordinates

本文提出一种在整数坐标平面点集中求解多对多匹配问题的亚二次精确算法，该算法在时间复杂度上优于传统方法，适用于点集在二维网格中的情况。

On flips in planar matchings

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph $\mathcal{G}_n$ has those matchings as vertices, and an edge between any two matchings that differ in…