Line and Plane Cover Numbers Revisited

Understand

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 $\pi^1_d(G)$.
• In 3D, the minimum number of \emph{planes} needed to cover all vertices of~$G$ is denoted by $\pi^2_3(G)$.
• When edges are also required to be covered, the corresponding numbers $\rho^1_d(G)$ and $\rho^2_3(G)$ are called the \emph{(strong) line cover number} and the \emph{(strong) plane cover number}.