2.5维中框的可见性表示
Visibility Representations of Boxes in 2.5 Dimensions
摘要：
我们开始研究2.5D盒可见性表示（2.5D-BR），其中顶点被映射到具有平面$z=0$中的底面的3D盒，并且边缘是平行于$x$ -或$y$ -轴的无障碍视线。我们证明：每个完全二部图允许一个2.5D-BR; $(i)$完全图$K_n$允许一个2.5D-BR当且仅当$n \leq 19$ ; $(iii)$每个路宽至多为$7$的图允许一个2.5D-BR，它可以在线性时间内计算。然后，我们将注意力转向2.5D网格框表示（2.5D-GBR），这是2. 5D-BR，使得每个框的底面是整数坐标处的单位正方形。我们证明了一个允许2.5D-GBR的$n$ -顶点图最多有$4n - 6 \sqrt{n}$边，并且这个界是紧的。最后，我们证明了判定给定图$G$是否允许具有给定足迹的2.5D-GBR是NP-完全的。2.5D-BR $Γ$的封装是$Γ$中盒子的底面集。
英文摘要：
We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $Γ$ is the set of bottom faces of the boxes in $Γ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
