凸点集上的几何图
Geometric graphs on convex point sets
摘要：
在本文中，我们引入了一类二部图，称为路径限制有序二部图，并将其作为凸点集上的单位距离图等已知几何图的抽象推广。在凸点集的框架下，我们还专注于广义版本的加布里埃尔图称为局部加布里埃尔图或$LGGs$。$LGGs$也可以看作是单位距离图的推广。路限制有序二部图也是$LGGs$的推广.研究了路限制有序二部图的一些结构性质，并证明了这类图具有$θ(n \log n)$的最大边复杂度。给出了凸点上的$UDGs$和$LGGs$有$O(n \log n)$边的另一种证明.
英文摘要：
In this note, we introduce a family of bipartite graphs called path restricted ordered bipartite graphs and present it as an abstract generalization of some well known geometric graphs like unit distance graphs on convex point sets. In the framework of convex point sets, we also focus on a generalized version of Gabriel graphs known as locally Gabriel graphs or $LGGs$. $LGGs$ can also be seen as the generalization of unit distance graphs. The path restricted ordered bipartite graph is also a generalization of $LGGs$. We study some structural properties of the path restricted ordered bipartite graphs and also show that such graphs have the maximum edge complexity of $θ(n \log n)$. It gives an alternate proof to the well known result that $UDGs$ and $LGGs$ on convex points have $O(n \log n)$ edges.
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