## Abstract

Given an n-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of O(n) steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.

Original language | English |
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Journal | S I A M Journal on Computing |

Volume | 46 |

Issue | 2 |

Pages (from-to) | 824-852 |

Number of pages | 29 |

ISSN | 0097-5397 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- INTERPOLATION
- TRIANGULATIONS
- morph
- planar graphs
- transformation