### 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 (US) |
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Pages (from-to) | 824-852 |

Number of pages | 29 |

Journal | SIAM Journal on Computing |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2017 |

Externally published | Yes |

### Keywords

- Morph
- Planar graphs
- Transformation

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*46*(2), 824-852. https://doi.org/10.1137/16M1069171