Title:

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as cliquewidth, and wellquasiorderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of cliquewidth immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study cliquewidth. As for wellquasiorderability, we conjecture that every hereditary graph class that is wellquasiordered by the induced subgraph relation also has bounded cliquewidth. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomialtime algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of cliquewidth, we describe a general framework for constructing bipartite graphs of large cliquewidth. As a consequence, we find a new minimal class of unbounded cliquewidth. We prove numerous positive and negative results regarding the wellquasiorderability of classes of bipartite graphs. This completes a characterisation of the wellquasiorderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine wellquasiorderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation.
