Monadic second-order definable graph orderings

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cases, we present combinatorial characterisations of when such a linear ordering is definable. In some cases, for instance for graph classes that omit a fixed graph as a minor, the presented conditions are necessary and sufficient; in other cases, they are only necessary. Other graph classes we consider include complete bipartite graphs, split graphs, chordal graphs, and cographs. We prove that orderability is decidable for the so called HR-equational classes of graphs, which are described by equation systems and generalize the context-free languages

Shrub-depth: Capturing Height of Dense Graphs

The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower ${\omega}$ levels of the MSO1 transduction hierarchy of simple graphs

Optimizing Tree Decompositions in MSO

The classic algorithm of Bodlaender and Kloks solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in MSO in the following sense: for every positive integer k, there is an MSO transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results, this implies that for every k there exists an MSO transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width

Definability equals recognizability for graphs of bounded treewidth

We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS 201