International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 472-486

# Pixel and Voxel Representations of Graphs

• Md. Jawaherul Alam
• Thomas Bläsius
• Ignaz Rutter
• Torsten Ueckerdt
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

## Abstract

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of.

We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for k-outerplanar graphs with n vertices, $$\varTheta (kn)$$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $$\varTheta (n^2)$$ voxels are always sufficient and sometimes necessary for any n-vertex graph. We improve this bound to $$\varTheta (n\cdot \tau )$$ for graphs of treewidth $$\tau$$ and to $$O((g+1)^2n\log ^2n)$$ for graphs of genus g. In particular, planar graphs admit representations with $$O(n\log ^2n)$$ voxels.

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Md. Jawaherul Alam
• 1
• Thomas Bläsius
• 2
• Ignaz Rutter
• 2
• Torsten Ueckerdt
• 2
• Alexander Wolff
• 3
1. 1.University of ArizonaTucsonUSA
2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany
3. 3.Universität WürzburgWürzburgGermany