### Abstract

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at most k new vertices such that each neighbor of v is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are 2-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify k-splittability in linear time, for a constant k.

Original language | English (US) |
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Pages (from-to) | 977-994 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2018 |

### Keywords

- Approximation
- Complete graphs
- Fixed-parameter tractable
- Genus-1 graphs
- Graph drawing
- Graph theory
- NP-hardness
- Planarity
- Splittable
- Thickness

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Algorithmica*,

*80*(3), 977-994. https://doi.org/10.1007/s00453-017-0328-y