再论线覆盖数与面覆盖数
Line and Plane Cover Numbers Revisited
摘要：
无直线交叉图的视觉复杂度的一个度量是覆盖所有顶点所需的最小线数。对于给定的图$G$，最小的这样的数目（在维度$d \in \{2,3\}$中的所有图上）被称为\emph{$d$-dimensional weak line cover number}，并且由$π^1_d(G)$表示。在3D中，覆盖~ $G$的所有顶点所需的\emph{planes}的最小数量由$π^2_3(G)$表示。当边缘也需要被覆盖时，相应的数字$ρ^1_d(G)$和$ρ^2_3(G)$被称为\emph{(strong) line cover number}和\emph{(strong) plane cover number}。计算这些覆盖数中的任何一个--除了$π^1_2(G)$ --都是NP困难的。Chaplick等人[WADS 2017]将计算$π^1_2(G)$的复杂性作为一个开放性问题提出。我们证明了对于给定的平面图~ $G$，判定是否有$π^1_2(G)=2$是NP-困难的.我们进一步证明了深度$d$，$G_d$的通用堆叠三角剖分具有$π^1_2(G_d)=d+1$ .对于~3D，我们证明了任何具有$ρ^2_3(G)=2$的$n$ -顶点图~ $G$至多有$5n-19$边，这是紧的.
英文摘要：
A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d \in \{2,3\}$) is called the \emph{$d$-dimensional weak line cover number} and denoted by $π^1_d(G)$. In 3D, the minimum number of \emph{planes} needed to cover all vertices of~$G$ is denoted by $π^2_3(G)$. When edges are also required to be covered, the corresponding numbers $ρ^1_d(G)$ and $ρ^2_3(G)$ are called the \emph{(strong) line cover number} and the \emph{(strong) plane cover number}. Computing any of these cover numbers -- except $π^1_2(G)$ -- is known to be NP-hard. The complexity of computing $π^1_2(G)$ was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph~$G$, whether $π^1_2(G)=2$. We further show that the universal stacked triangulation of depth~$d$, $G_d$, has $π^1_2(G_d)=d+1$. Concerning~3D, we show that any $n$-vertex graph~$G$ with $ρ^2_3(G)=2$ has at most $5n-19$ edges, which is tight.
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