Homogeneous sets in graphs and hypergraphs

A set of vertices in a graph or a hypergraph is called homogeneous if it is independent, that is it does not contain any edge, or if it is complete, that is it contains all possible pairs or subsets of it as edges. We investigate the properties of graphs and hypergraphs in two cases of imposed restrictions on the structure of their homogeneous sets. First we study the asymptotic structure of random perfect graphs. We give a generation model which yields such graphs almost uniformly, with an additive error of e-Ω(n) in the total variation distance. We use this model to determine a number of properties of random perfect graphs, including the distribution of the stability and the clique number, the probability of containing a fixed induced subgraph, Hamiltonicity, clique-colourability, connectivity, edge colouring, and the limit of a uniformly drawn sequence of perfect graphs. In the second part, we give a hypergraph parameter μ(H), called minor- matching number, with the property that hypergraphs H with bounded rank and minor-matching number contain a polynomially-bounded number of maximal independent sets. In the other direction, every hypergraph H contains at least 2μ(H) maximal independent sets. A number of hard hypergraph problems, including maximum-sized independent set, k-colouring and hypergraph homomorphism can be solved in polynomial time if a list with all maximal independent sets of the hypergraph is given as part of the input, and hence a family of instances with bounded minor-matching number of the input hypergraph form a new polynomial class for the problems above. The class can further be generalised by considering the maximum minor matching number of a bag in a tree decomposition as a new treewidth measure. We explain how to use this measure, defined as minor-matching treewidth, to solve hard problems and how to algorithmically construct a tree decomposition with approximate minimal width.