Lettericity is a graph parameter measuring how close a graph is to a word and geometric griddability measures how close a permutation is to a monotone permutation. Both of these parameters enjoy nice properties, in particular graphs of lettericity k are definable in Monadic second-order logic and permutations that are geometric M-griddability are in bijection with a regular language. I will first present both and some properties, and show that these seemingly unrelated parameters are in fact equivalent in permutation graphs, i.e., a permutation class has small geometric griddability iff the associated permutation graph class has small lettericity.

In the context of large-scale networks, fault tolerance is a necessity. Self-stabilization is an algorithmic approach to fault tolerance in distributed systems. Its aim is to manage processors' memory corruption due to transient failures. The efficiency of a self-stabilizing algorithm is characterized by several parameters, including (1) the time taken by the system to return to a legal configuration after an arbitrary corruption of its processors' memory, and (2) the memory space used by the processors to execute the algorithm. Minimizing memory space is motivated by many aspects, such as networks (e.g. sensor networks) with limited memory space, minimizing data exchanges between processors, and limiting information storage to use redundancy techniques. This seminar will present self-stabilization through two main classical problems: constructing spanning trees, especially those of minimum weight, and electing a leader. It will cover the links between self-stabilization and distributed decision techniques, including computing lower bounds on memory space needed to solve the aforementioned problems using self-stabilizing algorithms.

Courcelle's theorem, which states that all MSO-expressible properties can be decided in linear time on graphs of bounded treewidth, is one of the most celebrated results in parameterized complexity, because it implies the existence of linear-time FPT algorithms for a host of NP- hard problems. Unfortunately, the hidden constant implied by this theorem is a tower of exponentials whose height increases with each quantifier alternation in the formula.

In this talk we take a high-level view of this area of research and survey results that attempt to improve upon this performance by considering more restricted classes of inputs. This is known to be impossible, even if we restrict the input to graphs of treewidth 1 (trees) and only consider FO logic; or if we consider graphs of pathwidth 1 (caterpillars) and consider MSO logic. This would seem to indicate that Courcelle's theorem is “best possible”. Surprisingly, we discover that in the only remaining case which has so far been overlooked, namely FO logic on graphs of constant pathwidth, all known hardness results fail, because the problem becomes tractable with an elementary parameter dependence on the input formula. Our result generalizes previously known meta-theorems for the much more restricted parameter tree-depth.

Results based on a preprint available here: https://arxiv.org/abs/2210.09899