Clean Citation Style
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Alam, Md. Jawaherul; Bläsius, Thomas; Rutter, Ignaz; Ueckerdt, Torsten; Wolff, Alexander Pixel and Voxel Representations of Graphs. Graph Drawing (GD) 2015: 472486
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 minimumsize representations is NPcomplete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for \(k\)outerplanar graphs with \(n\) vertices, \(\Theta(kn)\) 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, \(\Theta(n^2)\) voxels are always sufficient and sometimes necessary for any \(n\)vertex graph. We improve this bound to \(\Theta(n \cdot \tau)\) for graphs of treewidth \(\tau\) and to \(O((g+1)^2 n \log^2 n)\) for graphs of genus \(g\). In particular, planar graphs admit representations with \(O(n\log^2 n)\) voxels.

Biedl, Therese C.; Bläsius, Thomas; Niedermann, Benjamin; Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz Using ILP/SAT to Determine Pathwidth, Visibility Representations, and other GridBased Graph Drawings. Graph Drawing (GD) 2013: 460471
We present a simple and versatile formulation of gridbased graph representation problems as an integer linear program (ILP) and a corresponding SAT instance. In a gridbased representation vertices and edges correspond to axisparallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problemspecific constraints. We describe a general \(d\)dimensional model for grid representation problems. This model can be used to solve a variety of NPhard graph problems, including pathwidth, bandwidth, optimum storientation, areaminimal (bark) visibility representation, boxicityk graphs and others. We implemented SATmodels for all of the above problems and evaluated them on the Rome graphs collection. The experiments show that our model successfully solves NPhard problems within few minutes on small to mediumsize Rome graphs.

Bläsius, Thomas; Karrer, Annette; Rutter, Ignaz Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices. Graph Drawing (GD) 2013: 220231
A simultaneous embedding (with fixed edges) of two graphs \(G^1\) and \(G^2\) with common graph \(G = G^1 \cap G^2\) is a pair of planar drawings of \(G^1\) and \(G^2\) that coincide on \(G\). It is an open question whether there is a polynomialtime algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe). In this paper, we present two results. First, a set of three lineartime preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph \(G^{\cup} = G ^1 \cup G^2\) , (2) most separating pairs of \(G^{\cup}\), and (3) connected components of \(G\) that are biconnected but not a cycle. Second, we give an \(O(n^3)\)time algorithm solving Sefe for instances with the following restriction. Let \(u\) be a pole of a \(P\)node \(\mu\) in the SPQRtree of a block of \(G^1\) or \(G^2\) . Then at most three virtual edges of \(\mu\) may contain common edges incident to \(u\). All algorithms extend to the sunflower case, i.e., to the case of more than three graphs pairwise intersecting in the same common graph.