The project studied various questions related to central problems in combinatorics, with the goal of gaining further insight into them. All of the questions could be motivated from algorithmic applications, but it is a project in pure mathematics of theoretical nature.
(1) Graph reconstruction. Reconstruction problems are everywhere in life: recovering a broken vase from its pieces, reconstructing the map of an ancient city from archeological findings, or reconstructing evolutionary histories from bacterial taxonomy data. It is very natural to ask when a graph (or graph property) can be reconstructed from its `pieces'. The main objective of the first work package is as follows.
(O1) Develop new techniques to reconstruct global graph parameters from local information.
Towards this, several explicit open questions were proposed, the first relating to finding the smallest size induced subgraphs for which you can reconstruct a tree. I am still involved in two ongoing collaborations related to this.
There are two projects that I did finish relating to this objective: one on reconstructing a graph from distance queries and the other on reconstructing a graph from the collection of k-tuples which form a connected graph. The latter has already been accepted by WG, an international conference in graph theory.
(2) Graph isomorphism: when are two networks, which may be described in very different ways, actually the same ('isomorphic')? The goal of this subpart of the project was to study a deeper mathematical theory that holds for much more general objects called manifolds, and see if the situation can be severely simplified in the special case of graphs.
(O2) Prove the analogous result of [G. Cornelissen and N. Peyerimhoff. Twisted isospectrality, homological wideness and isometry] for graphs: for a pair of cospectral graphs, give a simple algorithmic construction of finitely many matrices whose spectra determine whether or not the graphs are isomorphic.
I had several meetings with the supervisor and his co-author and indeed the situation seems to simplify quite a lot for graphs. This project is not yet finished and is still in the 'research' phase.
(3) Cerny's conjecture: what is the smallest length of a reset word, an instruction with the property that no matter where you start in a network, you always end up in the same location?
(O3) Improve our understanding of the smallest reset word of synchronising automata.
This has one part relating to what happens for a random automaton (3b) and one related to parameterized complexity (3a). For the random case, during the first month my project started, the upper bound got settled by an independent group of researchers (
https://arxiv.org/pdf/2207.14108.pdf(odnośnik otworzy się w nowym oknie)) with whom I then discussed the lower bound a bit. Before their work got put online, together with the two junior researchers I planned to work on this with during the proposal, we had obtained a weaker result but with simpler methods (I started working on this already before the start of the grant). I would have still written this down as a blogpost for this grant, since our methods were different than theirs, but before this could happen there was another researcher who 'scooped' us on this also.
We did gain more insight into the parameterized complexity of this problem and various relating problems, together with the second supervisor and a visitor.
