### 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) |
---|---|

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Algorithmica |

DOIs | |

State | Accepted/In press - Jun 2 2017 |

### Fingerprint

### 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

### Cite this

*Algorithmica*, 1-18. https://doi.org/10.1007/s00453-017-0328-y

**On the Planar Split Thickness of Graphs.** / Eppstein, David; Kindermann, Philipp; Kobourov, Stephen G; Liotta, Giuseppe; Lubiw, Anna; Maignan, Aude; Mondal, Debajyoti; Vosoughpour, Hamideh; Whitesides, Sue; Wismath, Stephen.

Research output: Contribution to journal › Article

*Algorithmica*, pp. 1-18. https://doi.org/10.1007/s00453-017-0328-y

}

TY - JOUR

T1 - On the Planar Split Thickness of Graphs

AU - Eppstein, David

AU - Kindermann, Philipp

AU - Kobourov, Stephen G

AU - Liotta, Giuseppe

AU - Lubiw, Anna

AU - Maignan, Aude

AU - Mondal, Debajyoti

AU - Vosoughpour, Hamideh

AU - Whitesides, Sue

AU - Wismath, Stephen

PY - 2017/6/2

Y1 - 2017/6/2

N2 - 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.

AB - 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.

KW - Approximation

KW - Complete graphs

KW - Fixed-parameter tractable

KW - Genus-1 graphs

KW - Graph drawing

KW - Graph theory

KW - NP-hardness

KW - Planarity

KW - Splittable

KW - Thickness

UR - http://www.scopus.com/inward/record.url?scp=85020136903&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020136903&partnerID=8YFLogxK

U2 - 10.1007/s00453-017-0328-y

DO - 10.1007/s00453-017-0328-y

M3 - Article

AN - SCOPUS:85020136903

SP - 1

EP - 18

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

ER -