## Colored Spanning Graphs for Set Visualization

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 (*ρ*/2 + 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.

keywords: Approximation, Computational Geometry, Fixed-Parameter Tractability, Graph Drawing, Graphs Theory

### Journal Article (peer-reviewed)

Akiyoshi Shioura, Bettina Speckmann, Ferran Hurtado, Maarten Löffler, Marc van Kreveld, Matias Korman, Rodrigo I. Silveira, Takeshi Tokuyama, Vera Sacristán

Colored Spanning Graphs for Set Visualization

Computational Geometry: Theory and Applications

68, 262–276, 2017

### Conference Proceedings (peer-reviewed)

Bettina Speckmann, Ferran Hurtado, Maarten Löffler, Marc van Kreveld, Matias Korman, Rodrigo I. Silveira, Vera Sacristán

Colored Spanning Graphs for Set Visualization

Proc. 21st International Symposium on Graph Drawing

LNCS 8242, 280–291, 2013

### Archived Publication (not reviewed)

Akiyoshi Shioura, Bettina Speckmann, Ferran Hurtado, Maarten Löffler, Marc van Kreveld, Matias Korman, Rodrigo I. Silveira, Takeshi Tokuyama, Vera Sacristán

Colored Spanning Graphs for Set Visualization

1603.00580, 2016

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