SWAT 2006: Algorithm Theory – SWAT 2006 pp 255-267

# Simultaneous Embedding with Two Bends per Edge in Polynomial Area

• Frank Kammer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

## Abstract

The simultaneous embedding problem is, given two planar graphs G 1=(V,E 1) and G 2=(V,E 2), to find planar embeddings ϕ(G 1) and ϕ(G 2) such that each vertex vV is mapped to the same point in ϕ(G 1) and in ϕ(G 2). This article presents a linear-time algorithm for the simultaneous embedding problem such that edges are drawn as polygonal chains with at most two bends and all vertices and all bends of the edges are placed on a grid of polynomial size. An extension of this problem with so-called fixed edges is also considered.

A further linear-time algorithm of this article solves the following problem: Given a planar graph G and a set of distinct points, find a planar embedding for G that maps each vertex to one of the given points. The solution presented also uses at most two bends per edge and a grid whose size is polynomial in the size of the grid that includes all given points. An example shows two bends per edge to be optimal.

## Keywords

Planar Graph Hamilton Path Dual Graph Outer Face Outerplanar Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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