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- W4287375737 abstract "A set of colored graphs are compatible, if for every color $i$, the number of vertices of color $i$ is the same in every graph. A simultaneous embedding of $k$ compatibly colored graphs, each with $n$ vertices, consists of $k$ planar polyline drawings of these graphs such that the vertices of the same color are mapped to a common set of vertex locations. We prove that simultaneous embedding of $kin o(log log n)$ colored planar graphs, each with $n$ vertices, can always be computed with a sublinear number of bends per edge. Specifically, we show an $O(min{c, n^{1-1/gamma}})$ upper bound on the number of bends per edge, where $gamma = 2^{lceil k/2 rceil}$ and $c$ is the total number of colors. Our bound, which results from a better analysis of a previously known algorithm [Durocher and Mondal, SIAM J. Discrete Math., 32(4), 2018], improves the bound for $k$, as well as the bend complexity by a factor of $sqrt{2}^{k}$. The algorithm can be generalized to obtain small universal point sets for colored graphs. We prove that $nlceil c/b rceil$ vertex locations, where $bge 1$, suffice to embed any set of compatibly colored $n$-vertex planar graphs with bend complexity $O(b)$, where $c$ is the number of colors." @default.
- W4287375737 created "2022-07-25" @default.
- W4287375737 creator A5015544517 @default.
- W4287375737 date "2021-01-17" @default.
- W4287375737 modified "2023-09-24" @default.
- W4287375737 title "Simultaneous Embedding of Colored Graphs" @default.
- W4287375737 doi "https://doi.org/10.48550/arxiv.2101.06596" @default.
- W4287375737 hasPublicationYear "2021" @default.
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