Pixel and Voxel Representations of Graphs

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, $\Theta(kn)$ 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 $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Shallow Minors, Graph Products and Beyond Planar Graphs

The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work is that many beyond planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of planar graphs, $k$-planar, $(k,p)$-cluster planar, fan-planar and $k$-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial $p$-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers. In addition, we show that $k$-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path

A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings

In this thesis, we study drawings of maximal outerplanar graphs that place vertices on integer coordinates. We introduce a new class of graphs, called umbrellas, and a new method of splitting maximal outerplanar graphs into systems of umbrellas. By doing so, we generate a new graph parameter, called the umbrella depth (ud), that can be used to approximate the optimal height of a drawing of a maximal outerplanar graph. We show that for any maximal outerplanar graph G, we can create a flat visibility representation of G with height at most 2ud(G) + 1. This drawing can be transformed into a straight-line drawing of the same height. We then prove that the height of any drawing of G is at least ud(G) + 1, which makes our result a 2-approximation for the optimal height. The best previously known approximation algorithm gave a 4-approximation. In addition, we provide an algorithm for finding the umbrella depth of G in linear time. Lastly, we compare the umbrella depth to other graph parameters such as the pathwidth and the rooted pathwidth, which have been used in the past for outerplanar graph drawing algorithms

New Parameters for Beyond-Planar Graphs

Parameters for graphs appear frequently throughout the history of research in this field. They represent very important measures for the properties of graphs and graph drawings, and are often a main criterion for their classification and their aesthetic perception. In this direction, we provide new results for the following graph parameters: – The segment complexity of trees; – the membership of graphs of bounded vertex degree to certain graph classes; – the maximal complete and complete bipartite graphs contained in certain graph classes beyond-planarity; – the crossing number of graphs; – edge densities for outer-gap-planar graphs and for bipartite gap-planar graphs with certain properties; – edge densities and inclusion relationships for 2-layer graphs, as well as characterizations for complete bipartite graphs in the 2-layer setting

Efficient Algorithms for Graph-Theoretic and Geometric Problems

This thesis studies several different algorithmic problems in graph theory and in geometry. The applications of the problems studied range from circuit design optimization to fast matrix multiplication. First, we study a graph-theoretical model of the so called ''firefighter problem''. The objective is to save as much as possible of an area by appropriately placing firefighters. We provide both new exact algorithms for the case of general graphs as well as approximation algorithms for the case of planar graphs. Next, we study drawing graphs within a given polygon in the plane. We present asymptotically tight upper and lower bounds for this problem Further, we study the problem of Subgraph Isormorphism, which amounts to decide if an input graph (pattern) is isomorphic to a subgraph of another input graph (host graph). We show several new bounds on the time complexity of detecting small pattern graphs. Among other things, we provide a new framework for detection by testing polynomials for non-identity with zero. Finally, we study the problem of partitioning a 3D histogram into a minimum number of 3D boxes and it's applications to efficient computation of matrix products for positive integer matrices. We provide an efficient approximation algorithm for the partitioning problem and several algorithms for integer matrix multiplication. The multiplication algorithms are explicitly or implicitly based on an interpretation of positive integer matrices as 3D histograms and their partitions