A parametric approach to hereditary classes

The “minimal class approach" consists of studying downwards-closed properties of hereditary graph classes (such as boundedness of a certain parameter within the class) by identifying the minimal obstructions to those properties. In this thesis, we look at various hereditary classes through this lens. In practice, this often amounts to analysing the structure of those classes by characterising boundedness of certain graph parameters within them. However, there is more to it than this: while adopting the minimal class viewpoint, we encounter a variety of interesting notions and problems { some more loosely related to the approach than others. The thesis compiles the author's work in the ensuing research directions

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

What is in# P and what is not?

For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum