This thesis deals with graphs having geometric representations. On one hand we consider graphs whose vertices can be mapped to geometric objets in an Euclidean space (for instance disks in the plane), such that two vertices are adjacent if and only if the corresponding objects intersect. Those are called "intersection graphs'', and if all objects are constrained to be, e.g. disks, then they are referred to as "disk graphs''. On the other hand we study graphs that can be represented in the plane such that vertices are mapped to points and edges to straight-line segments, such that no two edges cross. Those are called "plane geometric graphs''. In this thesis, we also have additional conditions. We consider for instance convex partitions, for which the union of the faces is equal to the convex hull of the points and each bounded face is convex. As a special case, we also study triangulations, for which we additionally require that every bounded face be a triangle.
For intersection graphs, we study the problem of finding a maximum clique in disk-like intersection graphs. In 1990, a seminal paper by Clark, Colbourn and Johnson showed that maximum clique can be solved in polynomial time in unit disk graphs. However, the complexity of maximum clique in disk graphs is still unknown. Recently, Bonamy et al.\ showed the existence of an EPTAS for maximum clique in disk graphs. This leads to the following questions: Are there superclasses of unit disk graphs in which maximum clique can be solved in polynomial time? Are there superclasses of disk graphs for which there is an EPTAS? Are there related classes for which we can show NP-hardness? Concerning the first question, we show that maximum clique can be solved in polynomial time in intersection graphs of translates of a fixed bounded convex set. Furthermore, we define a superclass C of both unit disk graphs and interval graphs, where C is defined as the intersection graph class of some specified sets, for which there exists a polynomial time algorithm. For the second question, we prove the existence of an EPTAS for homothets of a fixed bounded and centrally symmetric convex set. We also give partial results toward showing the existence of an EPTAS for intersection graphs of convex pseudo-disks. Finally, for the third question, we show that maximum clique is NP-hard, and even APX-hard, in intersection graphs of unit disks and axis-parallel rectangles.
Concerning triangulations and convex partitions, we study two problems that were previously only considered under the assumption that no three of the n input points are on a line. For triangulations, we extend a result by Wagner and Welzl and show that the bistellar flip graph is (n-3)-connected. For convex partitions, we provide the first approximation algorithms for computing convex partitions with as few faces as possibles when three points or more may lie on a line. In particular, we give an O(log(OPT))-approximation algorithm running in time O(n^8), where OPT denotes the size of a minimum solution. We additionally provide an O(sqrt(n)log(n))-approximation algorithm running in time O(n^2). We also show that minimising the number of faces is NP-hard
