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# Sparsifying Disk Intersection Graphs for Reliable Connectivity

$\newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}{#1} \newcommand{\Disks}{\Mh{\mathcal{C}}}% \newcommand{\BSet}{\Mh{B}}% \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$
Sariel Har-Peled, and Eliot W. Robson.
The intersection graph induced by a set $\Disks$ of $n$ disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of vertices. In this work, we present a sparsification algorithm that maintains connectivity between two disks in the computed graph, if the original graph remains well-connected'' even after removing an arbitrary attack'' set $\BSet \subseteq \Disks$ from both graphs. Thus, the new sparse graph has similar reliability to the original disk graph, and can withstand catastrophic failure of nodes while still providing a connectivity guarantee for the remaining graph. The new graph has near linear complexity, and can be constructed in near-linear time. The algorithm extends to any collection of shapes in the plane with near linear union complexity.
PDF. : arXiv.