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Eliminating Crossings in Ordered Graphs

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, Alexander Wolff

TL;DR

We investigate eliminating crossings in ordered graphs via fixed-vertex-order book embeddings and related edge-deletion and multi-track drawing variants. The authors deliver a fixed-parameter subexponential algorithm for Edge Deletion to 1-Page $d$-Planar with runtime $2^{O(d\sqrt{k}\log(d+k))} \cdot n^{O(1)}$, built from branching on highly-crossed edges and balanced-separator arguments that bound the conflict graph's treewidth. They also provide XP algorithms parameterized by the spine-hitting set size $h$ and the page count $p$, and extend the framework to spine+$t$-track drawings using subset-convolution DP and flow-based methods. An exact $2^m \cdot n^{O(1)}$-time algorithm computes the fixed-vertex-order page number, alongside $O(\log n)$-approximation and reductions to circle-graph independent-set problems for various page counts. Collectively, these results advance the parameterized complexity of crossing minimization in ordered graphs and offer practical tools for constrained graph drawing on fixed vertex orders.

Abstract

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with $n$ vertices and $m$ edges can be computed in $2^m \cdot n^{O(1)}$ time. An $O(\log n)$-approximation of this number can be computed efficiently. We can decide in $2^{O(d \sqrt{k} \log (d+k))} \cdot n^{O(1)}$ time whether it suffices to delete $k$ edges of an ordered graph to obtain a $d$-planar layout (where every edge crosses at most $d$ other edges) on one page. As an additional parameter, we consider the size $h$ of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For $h=1$, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number $p$. For $h>1$, we give an XP algorithm with respect to $h+p$. Finally, we consider spine+$t$-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of $t$ tracks, each of which is a straight line on a separate page, parallel to the spine.

Eliminating Crossings in Ordered Graphs

TL;DR

We investigate eliminating crossings in ordered graphs via fixed-vertex-order book embeddings and related edge-deletion and multi-track drawing variants. The authors deliver a fixed-parameter subexponential algorithm for Edge Deletion to 1-Page -Planar with runtime , built from branching on highly-crossed edges and balanced-separator arguments that bound the conflict graph's treewidth. They also provide XP algorithms parameterized by the spine-hitting set size and the page count , and extend the framework to spine+-track drawings using subset-convolution DP and flow-based methods. An exact -time algorithm computes the fixed-vertex-order page number, alongside -approximation and reductions to circle-graph independent-set problems for various page counts. Collectively, these results advance the parameterized complexity of crossing minimization in ordered graphs and offer practical tools for constrained graph drawing on fixed vertex orders.

Abstract

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with vertices and edges can be computed in time. An -approximation of this number can be computed efficiently. We can decide in time whether it suffices to delete edges of an ordered graph to obtain a -planar layout (where every edge crosses at most other edges) on one page. As an additional parameter, we consider the size of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For , we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number . For , we give an XP algorithm with respect to . Finally, we consider spine+-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of tracks, each of which is a straight line on a separate page, parallel to the spine.
Paper Structure (9 sections, 15 theorems, 5 figures, 1 table)

This paper contains 9 sections, 15 theorems, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Computing the Fixed-Vertex-Order Page Number
  3. Edge Deletion to 1-Page $d$-Planar
  4. Branching
  5. Balanced Separators in the Conflict Graph
  6. Proof of Theorem \ref{['thm:sub-exp']}
  7. Edge Deletion to $p$-Page Planar
  8. Multiple-Track Crossing Minimization
  9. Open Problems

Key Result

Theorem 1

Given an ordered graph $(G,\sigma)$ with $n$ vertices and $m$ edges, and a positive integer $p\xspace$, we can compute the values $\operatorname{cr}_1(G,\sigma),\dots,\operatorname{cr}_p\xspace(G,\sigma)$ in $2^m \cdot n^{\mathcal{O}(1)}$ time. In particular, given a budget $p$ of pages, we can comp

Figures (5)

  • Figure 1: A 3-page book embedding of $K_5$ with fixed vertex order. For each edge, we can choose on which page it is drawn. Note that $K_5$ cannot be drawn on two pages without crossings.
  • Figure 2: Case distinction for the proof of \ref{['lem:separator']}.
  • Figure 4: Encoding $\langle \mathcal{E}^1,\mathcal{E}^2 \rangle$ of a 2-page drawing for an instance with hitting set $H=\{a,b,c\}$ (red crosses). For each $X \subseteq H$ and page $q \in [2]$, the edges $e_X^q$ and $f_X^q$ (if they exist) are thicker than the other edges. Each colored region corresponds to a set of edges that bridge the same subset of $H$.
  • Figure 6: A spine+3-track drawing. In this example, $B_1$ has two vertices, $B_2$ has four vertices and $B_3$ has three vertices. The drawing has $2+5+2=9$ crossings.
  • Figure 7: Two different orders $\sigma_B$ give different number of crossings in the spine+$1$-track drawing: 10 on the left and 2 on the right.

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • lemma 1
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • lemma 2
  • lemma 3: Balanced Separator in the Conflict Graph
  • Theorem 6
  • lemma 4
  • ...and 5 more