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On $k$-Plane Insertion into Plane Drawings

Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, Ignaz Rutter

TL;DR

It is shown that the problem of plane Insertion into Plane drawing is NP-complete for every k-plane, even when G is biconnected and the set of edges forms a matching or a path.

Abstract

We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set $F$ of edges, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we show that the problem is NP-complete for every $k\ge 1$, even when $G$ is biconnected and the set $F$ of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that $k=1$ and $G$ is a triangulation.

On $k$-Plane Insertion into Plane Drawings

TL;DR

It is shown that the problem of plane Insertion into Plane drawing is NP-complete for every k-plane, even when G is biconnected and the set of edges forms a matching or a path.

Abstract

We introduce the -Plane Insertion into Plane drawing (-PIP) problem: given a plane drawing of a planar graph and a set of edges, insert the edges in into the drawing such that the resulting drawing is -plane. In this paper, we show that the problem is NP-complete for every , even when is biconnected and the set of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that and is a triangulation.
Paper Structure (7 sections, 6 theorems, 11 figures)

This paper contains 7 sections, 6 theorems, 11 figures.

Table of Contents

  1. Introduction
  2. $1$-PIP: Efficiently inserting edges into a triangulation
  3. $k$-PIP: Inserting a path or a matching is NP-complete
  4. The variable gadget.
  5. The clause gadget.
  6. Propagating the variable state.
  7. Conclusion

Key Result

Theorem 1

$1$-PIP can be solved in linear time for instances $(G,\Gamma,F)$ where $G$ is a triangulation.

Figures (11)

  • Figure 1: (a) The $1$-PIP problem: a plane graph $G$, a set $F$ of edges, and a 1-plane drawing of $G+F$. (b) In a triangulation, an edge in $G$ (bold) can only be an option for a single edge in $F$ (green) and clashes with at most four other options (blue).
  • Figure 3: Rectilinear representation of the variable-clause incidence graph of a Planar Monotone 3-SAT instance.
  • Figure 4: Different representations used in the drawings of our construction. (Left) every vertex and every edge, (middle) a simplification, and (right) a highly abstracted representation.
  • Figure 5: Drawing of the variable gadget illustrating \ref{['lem:variable']}.
  • Figure 6: Drawing of the clause gadget illustrating \ref{['lem:clause']}.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Corollary 6