### Abstract

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an (1/2ρ+1)- approximation, where ρ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.

Original language | English |
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Title of host publication | Graph Drawing - 21st International Symposium, GD 2013, Revised Selected Papers |

Pages | 280-291 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 2013 Dec 1 |

Externally published | Yes |

Event | 21st International Symposium on Graph Drawing, GD 2013 - Bordeaux, France Duration: 2013 Sep 23 → 2013 Sep 25 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8242 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 21st International Symposium on Graph Drawing, GD 2013 |
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Country | France |

City | Bordeaux |

Period | 13/9/23 → 13/9/25 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph Drawing - 21st International Symposium, GD 2013, Revised Selected Papers*(pp. 280-291). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8242 LNCS). https://doi.org/10.1007/978-3-319-03841-4-25