Line and Plane Cover Numbers Revisited

• Therese Biedl
• Stefan Felsner
• Henk Meijer
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph G, the minimum such number (over all drawings in dimension $$d \in \{2,3\}$$) is called the d-dimensional weak line cover number and denoted by $$\pi ^1_d(G)$$. In 3D, the minimum number of planes needed to cover all vertices of G is denoted by $$\pi ^2_3(G)$$. When edges are also required to be covered, the corresponding numbers $$\rho ^1_d(G)$$ and $$\rho ^2_3(G)$$ are called the (strong) line cover number and the (strong) plane cover number.

Computing any of these cover numbers—except $$\pi ^1_2(G)$$—is known to be NP-hard. The complexity of computing $$\pi ^1_2(G)$$ was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph G, whether $$\pi ^1_2(G)=2$$. We further show that the universal stacked triangulation of depth d, $$G_d$$, has $$\pi ^1_2(G_d)=d+1$$. Concerning 3D, we show that any n-vertex graph G with $$\rho ^2_3(G)=2$$ has at most $$5n-19$$ edges, which is tight.

Notes

Acknowledgments

This research started at the Bertinoro Workshop on Graph Drawing 2017. We thank the organizers and other participants, in particular Will Evans, Sylvain Lazard, Pavel Valtr, Sue Whitesides, and Steve Wismath. We also thank Alex Pilz and Piotr Micek for enlightening conversations.

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Authors and Affiliations

2. 2.TU BerlinBerlinGermany
3. 3.University College RooseveltMiddelburgThe Netherlands
4. 4.Universität WürzburgWürzburgGermany