We call a graph G separable if a balanced separator can be computed for G of size O(n^ε) with ε < 1. Many real-world graphs are separable such as graphs of bounded genus, graphs of constant treewidth, and graphs excluding a fixed minor. In particular, the well-known planar graphs are separable. We present a succinct encoding of separable graphs G such that, after the encoding is computed, any number of depth-first searches (DFS) can be performed from any given start vertex, each in o(n) time and o(n) bits in the word RAM model. After the execution of a DFS, the succinct encoding of G is augmented such that the DFS tree is encoded inside the encoding while maintaining succinctness. Afterward, the encoding provides common DFS-related queries in constant time. These queries include queries such as lowest-common ancestor of two given vertices in the DFS tree or queries that output the lowpoint of a given vertex in the DFS tree. Furthermore, for planar graphs, we show that the succinct encoding can be computed in O(n) bits and expected linear time, and a compact variant can be constructed in O(n) time and bits. For other separable graph classes 𝒢 the runtime and space usage depends on the specific algorithms used to find balanced separators in graphs of 𝒢.

We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on structures of bounded tree-depth. Order-invariance is undecidable in general and, thus, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on structures of bounded tree-depth, order-invariant FO has the same expressive power as FO. Our proof technique allows for a fine-grained analysis of the succinctness of this translation. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. We obtain similar results for MSO. It is known that the expressive power of MSO and FO coincide on structures of bounded tree-depth. We provide a translation from MSO to FO and we show that this translation is essentially optimal regarding the formula size. As a further result, we show that order-invariant MSO has the same expressive power as FO with modulo-counting quantifiers on bounded tree-depth structures.

Monadic-second order logic (MSO-logic) is successfully applied in both language theory and algorithm design. In the former, properties definable by MSO-formulas are exactly the regular properties on many structures like, most prominently, strings. In the latter, solving a problem for structures of bounded tree width is routinely done by defining it in terms of an MSO-formula and applying general formula-evaluation procedures like Courcelle's. The present paper furthers the study of second-order logics with close connections to language theory and algorithm design beyond MSO-logic. We introduce a logic that allows to expand a given structure with an existentially quantified tree decomposition of bounded width and test an MSO-definable property for the resulting expanded structure. It is proposed as a candidate for capturing the notion of "context-free graph properties" since it corresponds to the context-free languages on strings, has the same closure properties, and an alternative definition similar to the one of Chomsky and Schützenberger for context-free languages. Besides studying its language-theoretic aspects, we consider its expressive power as well as the algorithmics of its satisfiability and evaluation problems.

Order-invariant formulas access an ordering on a structure's universe, but the model relation is independent of the used ordering. They are frequently used for logic-based approaches in computer science. Order-invariant formulas capture unordered problems of complexity classes and they model the independence of the answer to a database query from low-level aspects of databases. We study the expressive power of order-invariant monadic second-order (MSO) and first-order (FO) logic on restricted classes of structures that admit certain forms of tree decompositions (not necessarily of bounded width).

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized in deterministic logarithmic space (logspace). This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous classes of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.

We study on which classes of graphs first-order logic ( fo ) and monadic second-order logic ( mso ) have the same expressive power. We show that for all classes C of graphs that are closed under taking subgraphs, fo and mso have the same expressive power on C if and only if, C has bounded tree depth. Tree depth is a graph invariant that measures the similarity of a graph to a star in a similar way that tree width measures the similarity of a graph to a tree. For classes just closed under taking induced subgraphs, we show an analogous result for guarded second-order logic ( gso ), the variant of mso that not only allows quantification over vertex sets but also over edge sets. A key tool in our proof is a Feferman--Vaught-type theorem that works for infinite collections of structures despite being constructive.

We study the complexity of model checking formulas in first-order logic parameterized by the number of distinct variables in the formula. This problem, which is not known to be fixed-parameter tractable, resisted to be properly classified in the context of parameterized complexity. We show that it is complete for a newly-defined complexity class that we propose as an analog of the classical class PSPACE in parameterized complexity. We support this intuition by the following findings: First, the proposed class admits a definition in terms of alternating Turing machines in a similar way as PSPACE can be defined in terms of polynomial-time alternating machines. Second, we show that parameterized versions of other PSPACE-complete problems, like winning certain pebble games and finding restricted resolution refutations, are complete for this class.

Graph embeddings of bounded Euler genus (that means, embeddings with bounded orientable or nonorientable genus) help to design time-efficient algorithms for many graph problems. Since linear-time algorithms are known to compute embeddings of any bounded Euler genus, one can always assume to work with embedded graphs and, thus, obtain fast algorithms for many problems on any class of graphs of bounded Euler genus.

We study on which classes of graphs first-order logic (FO) and monadic second-order logic (MSO) have the same expressive power. We show that for each class of graphs that is closed under taking subgraphs, FO and MSO have the same expressive power on the class if, and only if, it has bounded tree depth. Tree depth is a graph invariant that measures the similarity of a graph to a star in a similar way that tree width measures the similarity of a graph to a tree. For classes just closed under taking induced subgraphs, we show an analogous result for guarded second-order logic (GSO), the variant of MSO that not only allows quantification over vertex sets but also over edge sets. A key tool in our proof is a Feferman-Vaught-type theorem that is constructive and still works for unbounded partitions.

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that MSO-definable problems are (a) solvable by uniform constant-depth circuit families (AC0 for decision problems and TC0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC1 for decision problems and #NC1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC0-completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC1. Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata's computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC0, which encapsulates most of the technical difficulties surrounding earlier results connecting tree automata and NC1.

Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when "linear time" is replaced by "logarithmic space." The transfer of the powerful theoretical framework of monadic second-order logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the log space world.