A fast algorithm for computing a planar support for non-piercing rectangles
<|reference_start|>A fast algorithm for computing a planar support for non-piercing rectangles: For a hypergraph $\mathcal{H}=(X,\mathcal{E})$ a \emph{support} is a graph $G$ on $X$ such that for each $E\in\mathcal{E}$, the induced subgraph of $G$ on the elements in $E$ is connected. If $G$ is planar, we call it a planar support. A set of axis parallel rectangles $\mathcal{R}$ forms a non-piercing family if for any $R_1, R_2 \in \mathcal{R}$, $R_1 \setminus R_2$ is connected. Given a set $P$ of $n$ points in $\mathbb{R}^2$ and a set $\mathcal{R}$ of $m$ \emph{non-piercing} axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph $(P,\mathcal{R})$ in $O(n\log^2 n + (n+m)\log m)$ time, where each $R\in\mathcal{R}$ defines a hyperedge consisting of all points of $P$ contained in~$R$. We use this result to show that if for a family of axis-parallel rectangles, any point in the plane is contained in at most $k$ pairwise \emph{crossing} rectangles (a pair of intersecting rectangles such that neither contains a corner of the other is called a crossing pair of rectangles), then we can obtain a support as the union of $k$ planar graphs.<|reference_end|>
@article{pal2024a,
    title={A fast algorithm for computing a planar support for non-piercing
  rectangles},
    author={Ambar Pal, Rajiv Raman, Saurabh Ray, Karamjeet Singh},
    journal={arXiv preprint arXiv:2410.02449},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.02449},
    primaryClass={cs.CG}
}