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Outer-(ap)RAC Graphs

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

An 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 outer-apRAC (ap for 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.

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Notes

  1. 1.

    Originally, the slopes where defined as 0 and \(\infty \) in [3]. We use the rotated version \(\pm 1\) which will allow us to simplify our discussion in Sect. 5.

  2. 2.

    In the literature there exists another recursive definition for series-parallel graphs [26]. However, one can obtain biconnectivity by a parallel composition with a single edge.

  3. 3.

    When considering the edges incident to \(v_i\) in clockwise order starting from the direction in which we reach \(v_i\) along O(B) coming from \(v_{i-1}\).

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Author information

Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Henry Förster & Julia Katheder

  2. Campus Heilbronn, Technische Universität München, Heilbronn, Germany

    Henry Förster

  3. Department of Engineering, University of Perugia, Perugia, Italy

    Giacomo Ortali

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  1. Henry Förster
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  2. Julia Katheder
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Correspondence to Henry Förster .

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Editors and Affiliations

  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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Förster, H., Katheder, J., Ortali, G. (2025). Outer-(ap)RAC Graphs. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_21

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