论文标题
边缘最小饱和K平面图
Edge-Minimum Saturated k-Planar Drawings
论文作者
论文摘要
对于一类$ \ Mathcal {d} $在飞机上无环（多）图的图纸，当在\ Mathcal {d} $中添加$ d \是\ emph {ata atavation}时，当$ d $添加到$ d $结果中，$ d $ d'在$ d'\ notin \ notin \ notin \ nation \ nation frust a prange in a prange in a prange a prange a a prange a a prange a satus a a a prapent a a sater a Amaloge a a a a rage a a a prange（和Erdős，Hajnal和Moon（1964）。我们专注于$ k $ - planar图纸，也就是说，在每个边缘的飞机上绘制的图形最多$ k $ times，以及所有$ k $ - $ - $ - planar绘图的$ \ nathcal {d} $遵守了许多限制，例如没有交叉事件，没有比一次或不一次的交叉或任何边缘交叉或任何越过的越野。虽然饱和的$ K $ - 平面图是几项先前作品的重点，但对这些图的稀疏程度的紧密界限尚不清楚。我们建立了一个通用框架，以确定所有自然类中所有$ n $ vertex饱和$ k $ - 平面图中的最小边数。例如，当允许发生事件交叉点，多次旋转和自划线时，最稀少的$ n $ vertex饱和$ k $ -pplanar图纸具有$ \ frac {2} {k - （k \ bmod 2）}}（n-1）（n-1）$ edges $ edges $ geq \ geq \ geq \ geq \ geq \ geq 4 $ $ \ frac {2（k+1）} {k（k-1）}（n-1）$ edge for任何$ k \ geq 6 $。
For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs in a graph class as introduced by Turán (1941) and Erdős, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $\mathcal{D}$ of all $k$-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated $k$-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all $n$-vertex saturated $k$-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest $n$-vertex saturated $k$-planar drawings have $\frac{2}{k - (k \bmod 2)} (n-1)$ edges for any $k \geq 4$, while if all that is forbidden, the sparsest such drawings have $\frac{2(k+1)}{k(k-1)}(n-1)$ edges for any $k \geq 6$.
