关于顶点邻近图和空层邻近图
On Vertex- and Empty-Ply Proximity Drawings
摘要：
我们开始研究直线图的顶点铺层，作为最近引入的铺层数的放松。考虑以每个顶点为中心的圆盘，其半径等于与顶点相关联的最长边的长度的一半。图形的顶点层数由最大圆盘数所覆盖的顶点决定。考虑这种放松的主要动机是将铺层的概念与接近图联系起来。事实上，如果我们将圆盘的集合解释为邻近区域，则顶点层数为1的绘图可以被视为弱邻近绘图，我们称之为空层绘图。我们展示了非平凡的层数和顶点层数之间的关系。然后，我们重点讨论了空层图，证明了一些性质，并研究了什么样的图允许空层图。最后证明了平面图的铺层和顶点铺层的一个下界。
英文摘要：
We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.
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