Uniqueness and complexity in generalised colouring

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: • Additive induced-hereditary families are uniquely factorisable into irreducible families. • If P and Q are additive induced-hereditary graph families, then (P,Q)colouring is NP-hard, with the exception of graph 2-colouring. Moreover, with the same exception, (P,Q)-colouring is NP-complete iff Pand Q-recognition are both in NP. This proves a 1997 conjecture of Kratochv́ıl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: • a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and vice versa; • extensions of Mihók’s construction of uniquely colourable graphs, and Scheinerman’s characterisations of compositivity, to disjoint compositive properties; • an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; • if G is a generating set for A ◦ B, where A and B are indiscompositive, then we can extract generating sets for A and B using a “greedy algorithm”.