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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.17565 |
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| _version_ | 1866909405620994048 |
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| author | Förster Katheder, Julia Ortali, Giacomo |
| author_facet | Förster Katheder, Julia Ortali, Giacomo |
| contents | An \emph{outer-RAC drawing} of a graph is a straight-line drawing where all vertices are incident to the outer cell and all edge crossings occur at a right angle. If additionally, all crossing edges are either horizontal or vertical, we call the drawing \emph{outer-apRAC} (\emph{ap} for \emph{axis-parallel)}. A graph is outer-(ap)RAC if it admits an outer-(ap)RAC drawing. We investigate the class of outer-(ap)RAC graphs. We show that the outer-RAC graphs are a proper subset of~the planar graphs with at most $2.5n-4$ edges where $n$ is the number of vertices. This density bound is tight, even for outer-apRAC graphs. Moreover, we provide an SPQR-tree based linear-time algorithm which computes an outer-RAC drawing for every given series-parallel graph of maximum degree four. As a complementing result, we present planar graphs of maximum degree four and series-parallel graphs of maximum degree five that are not outer-RAC. Finally, for series-parallel graphs of maximum degree three we show how to compute an outer-apRAC drawing in linear time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17565 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Outer-(ap)RAC Graphs Förster Katheder, Julia Ortali, Giacomo Computational Complexity An \emph{outer-RAC drawing} of a graph is a straight-line drawing where all vertices are incident to the outer cell and all edge crossings occur at a right angle. If additionally, all crossing edges are either horizontal or vertical, we call the drawing \emph{outer-apRAC} (\emph{ap} for \emph{axis-parallel)}. A graph is outer-(ap)RAC if it admits an outer-(ap)RAC drawing. We investigate the class of outer-(ap)RAC graphs. We show that the outer-RAC graphs are a proper subset of~the planar graphs with at most $2.5n-4$ edges where $n$ is the number of vertices. This density bound is tight, even for outer-apRAC graphs. Moreover, we provide an SPQR-tree based linear-time algorithm which computes an outer-RAC drawing for every given series-parallel graph of maximum degree four. As a complementing result, we present planar graphs of maximum degree four and series-parallel graphs of maximum degree five that are not outer-RAC. Finally, for series-parallel graphs of maximum degree three we show how to compute an outer-apRAC drawing in linear time. |
| title | Outer-(ap)RAC Graphs |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2411.17565 |