Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Minimum Bends in a Polyline Drawing with Fixed Vertex Locations   We consider embeddings of planar graphs in $R^2$ where vertices map to points
and edges map to polylines. We refer to such an embedding as a polyline
drawing, and ask how few bends are required to form such a drawing for an
arbitrary planar graph. It has long been known that even when the vertex
locations are completely fixed, a planar graph admits a polyline drawing where
edges bend a total of $O(n^2)$ times. Our results show that this number of
bends is optimal. In particular, we show that $\Omega(n^2)$ total bends is
required to form a polyline drawing on any set of fixed vertex locations for
almost all planar graphs. This result generalizes all previously known lower
bounds, which only applied to convex point sets, and settles 2 open problems.

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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th