Random graphs with structural constraints

Abstract

The classical way to measure the performance of an algorithm is to consider its worst-case performance. However, in practice this performance measure is often overly pessimistic. This stems from the fact that the worst-case analysis is frequently based on few inputs, which might have a quite artificial structure that does not (or seldom) appear as part of a “typical” input instance. A natural alternative is to consider the average-case performance of an algorithm, that is, we analyse the performance of an algorithm assuming that the input instances are drawn from all possible instances according to a given probability distribution. In order to analyse the average-case performance of an algorithm and to design algorithms with better average-case performance, it is crucial to understand the properties of a “typical” input instance. In many real-world scenarios – for example in chip manufacturing and drawing of diagrams – one has to deal with algorithms that take graphs with structural constraints, such as planar graphs, as input. Investigating properties of such constrained graph classes and developing new tools and methods, which help to cope with the difficulty of the dependence of the edges, are central to advance the state of the art in this area of research. In this thesis, we focus on such constrained graph classes, namely planar graphs with given average degree, cactus graphs, block graphs, and (maximal) K3,3-minor-free graphs. We are interested to prove that graphs on n nodes, drawn uniformly at random from the set of all graphs on n nodes of a class with structural constraints, have specific properties with high probability (w.h.p., i.e., with probability tending to 1 as n → ∞). We apply different techniques and provide a new method to obtain results about several graph classes with structural constraints. Our first result is about planar graphs. Planar graphs are well-known and well-studied combinatorial objects in graph theory. Roughly speaking, a graph is planar if it can be drawn in the plane in such a way that no two edges cross. A random planar graph Rn is drawn uniformly at random from the set P(n) of all simple labelled planar graphs on the node set {1, . . . , n}. Here we consider planar graphs with given average degree. More precisely, we are interested in properties of a random planar graph Rn,q which is drawn uniformly at random from the class P(n, bqnc) of simple labelled planar graphs with n nodes and bqnc edges, where 1 < q < 3 and the average degree is about 2q. We exploit proof techniques of McDiarmid, Steger, and Welsh [MSW05, MSW06], to show that for all 1 < q < 3 the random planar graphRn,q has properties similar to those of a random planar graph Rn. For example, we show that Rn,q contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint copies of each given fixed connected planar