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Outer-(ap)RAC Graphs

Förster, Julia Katheder, Giacomo Ortali

Abstract

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.

Outer-(ap)RAC Graphs

Abstract

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 edges where 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.

Paper Structure

This paper contains 19 sections, 15 theorems, 5 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Preliminaries
  4. Topological Results
  5. Obstructions for Outer-RAC Graphs
  6. Outer-RAC Drawings for Bounded Degree SP-Graphs
  7. Initialization.
  8. $\mu$ is a $Q$-node.
  9. $\mu$ is a $P$-node.
  10. $\mu$ is a $S$-node.
  11. Running time.
  12. Open Problems
  13. Proof of \ref{['thm:density-upperbound']}
  14. Required Crossings in Skeletons of S-nodes in Outer-apRAC Drawings
  15. Outer-RAC Drawings of Subquartic SP-Graphs
  16. ...and 4 more sections

Key Result

lemma 1

Let $G$ be an SP-graph with $n \geq 3$ vertices and let $\mathcal{T}$ be its SQPR-tree rooted at any $Q$-node. Then, $\mathcal{T}$ contains an $S$-node $\mu$ such that all children of $\mu$ are $Q$-nodes.

Figures (11)

  • Figure 1: (a) An outer RAC graph with its crossing edges being decomposed into blocks (colored edges) and the outlines of the blocks (colored regions). (b) Illustration for the proof of \ref{['thm:planar']}.
  • Figure 2: Illustrations for the proof of (a) \ref{['thm:density-lowerbound']} (b) \ref{['lem:two-parallel-short-paths']} and (c) \ref{['lem:three-parallel-short-paths']}.
  • Figure 3: (a) Invariants \ref{['inv:sc-ve-vertical']} and \ref{['inv:sc-res-region']}. (b) and (c) initialization of \ref{['thm:sp-degree-3']}.
  • Figure 4: (a - c) Treatment of $P$-nodes in the algorithm in the proof of \ref{['thm:sp-degree-3']}. (d) Treatment of $S$-nodes in the algorithm in the proof of \ref{['thm:sp-degree-3']}.
  • Figure 5: (a) Graph $L_9 \in \mathcal{L}$. (b) Outer-RAC drawing of $L_9$. (c) Drawing of a $9$-cycle as described in the proof of \ref{['lem:not-nice-ap']}. (d) Outer-apRAC drawing of $L_9$. (e) An outer-apRAC SP-graph with maximum degree $6$.
  • ...and 6 more figures

Theorems & Definitions (31)

  • lemma 1
  • proof
  • corollary 1
  • theorem 1
  • proof
  • lemma 2
  • proof
  • theorem 2
  • proof : Sketch of Proof
  • theorem 3: DBLP:journals/algorithmica/AuerBBGHNR16DBLP:journals/ijcga/DehkordiE12DBLP:journals/ipl/Didimo13
  • ...and 21 more