6 research outputs found
Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes
Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth.
Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class C of graphs is flip-flat if for every fixed radius r, every sufficiently large set of vertices of a graph G ? C contains a large subset of vertices with mutual distance larger than r, where the distance is measured in some graph G\u27 that can be obtained from G by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms
Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures
We introduce a new data structure for answering connectivity queries in
undirected graphs subject to batched vertex failures. Precisely, given any
graph G and integer k, we can in fixed-parameter time construct a data
structure that can later be used to answer queries of the form: ``are vertices
s and t connected via a path that avoids vertices ?'' in time
. In the terminology of the literature on data structures, this
gives the first deterministic data structure for connectivity under vertex
failures where for every fixed number of failures, all operations can be
performed in constant time.
With the aim to understand the power and the limitations of our new
techniques, we prove an algorithmic meta theorem for the recently introduced
separator logic, which extends first-order logic with atoms for connectivity
under vertex failures. We prove that the model-checking problem for separator
logic is fixed-parameter tractable on every class of graphs that exclude a
fixed topological minor. We also show a weak converse. This implies that from
the point of view of parameterized complexity, under standard complexity
assumptions, the frontier of tractability of separator logic is almost exactly
delimited by classes excluding a fixed topological minor.
The backbone of our proof relies on a decomposition theorem of Cygan et al.
[SICOMP '19], which provides a tree decomposition of a given graph into bags
that are unbreakable. Crucially, unbreakability allows to reduce separator
logic to plain first-order logic within each bag individually. We design our
model-checking algorithm using dynamic programming over the tree decomposition,
where the transition at each bag amounts to running a suitable model-checking
subprocedure for plain first-order logic. This approach is robust enough to
provide also efficient enumeration of queries expressed in separator logic