arXiv subjects

Bruno Courcelle

Publications and source records attributed to Bruno Courcelle.

At least 19 recordsLinked to original sources

arXiv2026

On first-order definable operations on relational structures

We survey the definitions and main properties of first-order (FO) definable unary operations on relational structures, called FO-transductions, and of FO-definable binary operations based on disjoint union and Cartesian product. We focus our study on Backwards Translation Theorems and Splitting Theorems that permit to express FO properties of output structures in terms of finitely many FO properties of the corresponding input ones. In the particular cases where the operations are defined by quantifier-free (QF) formulas, the quantifier-heights of the obtained sentences are no larger than those of the input ones. It follows that the class of finite models of a FO sentence is recognizable with respect to the considered QF operations. Recognizability has interesting algorithmic properties based on finite automata on terms, for structures having bounded tree-width or clique-width. We extend our results to FO sentences constructed with modulo counting existential quantifiers.

cs.LO↗

arXiv2025

On describing trees and quasi-trees from their leaves

Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees, we define relational structures on their leaves that characterize them up to isomorphism. These structures have characterizations by universal first-order sentences. Furthermore, we consider cases where O-trees and quasi-trees can be reconstructed from their leaves by CMSO-transductions. These transductions are transformations of relational structures defined by monadic second-order (MSO) formulas. The letter "C" for counting refers to the use of set predicates that count cardinalities of finite sets modulo fixed integers. O-trees and quasi-trees make it possible to define respectively, the modular decomposition and the rank-width of a countable graph. Their constructions from their leaves by transductions of different types apply to rank-decompositions, and to modular decomposition and to other canonical graph decompositions.

cs.LO↗

arXiv2022

Unfoldings and coverings of weighted graphs

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings. We prove similar statements for unfoldings. Our study of the difficult proof of Leighton's Theorem lead us to generalize coverings and similarly, unfoldings, by attaching finite or infinite weights to edges of the covered or unfolded graphs. This generalization yields a canonical factorization of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing infinite weights provides us with finite descriptions of regular trees having nodes of countably infinite degree. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

cs.LO↗

arXiv2021

Induced betweenness in order-theoretic trees

The ternary relation $B(x,y,z)$ of betweenness states that an element $y$ is between the elements $x$ and $z$, in some sense depending on the considered structure. In a partially ordered set $(N,\leq)$, $B(x,y,z):\Longleftrightarrow x<y<z\vee z<y<x$. The corresponding betweenness structure is $(N,B)$. The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, we define $B(x,y,z)$ to mean that $x<y<z$ or $z<y<x$ or $x<y\leq x\sqcup z$ or $z<y\leq x\sqcup z$ provided the least upper-bound $x\sqcup z$ of $x$ and $z$ is defined when $x$ and $z$ are incomparable. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.

cs.LO↗

arXiv2021

Order-theoretic trees: monadic second-order descriptions and regularity

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any countable linear order. Such generalized infinite trees yield convenient definitions of the rank-width and the modular decomposition of countable graphs. We define an algebra based on only four operations that generate up to isomorphism and via infinite terms these order-theoretic trees and forests. We prove that the associated regular objects, those defined by regular terms, are exactly the ones that are the unique models of monadic second-order sentences.

cs.LO↗

arXiv2020

A unified algorithm for colouring graphs of bounded clique-width

Clique-width is one of the graph complexity measures leading to polynomial special-case algorithms for generally NP-complete problems, e.g. graph colourability. The best two currently known algorithms for verifying c-colourability of graphs represented as clique-width terms are optimised towards two different extreme cases, a constant number of colours and a very large number of colours. We present a way to unify these approaches in a single relatively simple algorithm that achieves the state of the art complexity in both cases. The unified algorithm also provides a speed-up for a large number of colours.

cs.CC↗

arXiv2020

Betweenness of partial orders

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the be-tweenness of a partial order.

cs.LO↗

arXiv2020

Axiomatization of betweenness in order-theoretic trees

The ternary betweenness relation of a tree, B(x,y,z) expresses that y is on the unique path between x and z. This notion can be extended to order-theoretic trees defined as partial orders such that the set of nodes larger than any node is linearly ordered. In such generalized trees, the unique "path" between two nodes can have infinitely many nodes. We generalize some results obtained in a previous article for the betweenness of join-trees. Join-trees are order-theoretic trees such that any two nodes have a least upper-bound. The motivation was to define conveniently the rank-width of a countable graph. We called quasi-tree the structure based on the betweenness relation of a join-tree. We proved that quasi-trees are axiomatized by a first-order sentence. Here, we obtain a monadic second-order axiomatization of betweenness in order-theoretic trees. We also define and compare several induced betweenness relations, i.e., restrictions to sets of nodes of the betweenness relations in generalized trees of different kinds. We prove that induced betweenness in quasi-trees is characterized by a first-order sentence. The proof uses order-theoretic trees.

cs.LO↗

arXiv2016

Algebraic and logical descriptions of generalized trees

Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound of the rank-widths of its finite induced subgraphs. Join-trees are the corresponding directed trees. They are useful to define the modular decomposition of a countable graph. We also consider ordered join-trees, that generalize rooted trees equipped with a linear order on the set of sons of each node. We define algebras with finitely many operations that generate (via infinite terms) these generalized trees. We prove that the associated regular objects (those defined by regular terms) are exactly the ones that are the unique models of monadic second-order sentences. These results use and generalize a similar result by W. Thomas for countable linear orders.

cs.LO↗

arXiv2015

Fly-automata for checking MSO 2 graph properties

A more descriptive but too long title would be : Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute "on the fly" the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quan-tifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.

cs.LO↗

arXiv2014

Fly-automata, model-checking and recognizability

The Recognizability Theorem states that if a set of finite graphs is definable by a monadic second-order (MSO) sentence, then it is recognizable with respect to the graph algebra upon which the definition of clique-width is based. Recognizability is an algebraic notion, defined in terms of congruences that can also be formulated by means of finite automata on the terms that describe the considered graphs. This theorem entails that the verification of MSO graph properties, or equivalently, the model-checking problem for MSO logic over finite binary relational structures, is fixed-parameter tractable (FPT) for the parameter consisting of the formula that expresses the property and the clique-width (or the tree-width) of the input graph or structure. The corresponding algorithms can be implemented by means of fly-automata whose transitions are computed on the fly and not tabulated. We review two versions of recognizability, we present fly-automata by means of examples showing that they can also compute values attached to graphs. We show that fly-automata with infinite sets of states yield a simple proof of the strong version of the Recognizability Theorem. This proof has not been published previously.

cs.LO↗

arXiv2013

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.

cs.LO↗

arXiv2013

Clique-width and edge contraction

We prove that edge contractions do not preserve the property that a set of graphs has bounded clique-width. This property is preserved by contractions of edges, one end of which is a vertex of degree 2.

cs.DM↗

arXiv2013

Computations by fly-automata beyond monadic second-order logic

We present logically based methods for constructing XP and FPT graph algorithms, parametrized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make possible to check properties that are not monadic second-order expressible because their states may include counters, so that their sets of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. Rather than new algorithmic results we present tools for constructing easily certain dynamic programming algorithms by combining predefined automata for basic functions and properties.

cs.LO↗

arXiv2010

On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids.

math.LO↗

arXiv2008

Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries.

cs.DS↗

arXiv2007

A multivariate interlace polynomial

We define a multivariate polynomial that generalizes several interlace polynomials defined by Arratia, Bollobas and Sorkin on the one hand, and Aigner and van der Holst on the other. We follow the route traced by Sokal, who defined a multivariate generalization of Tutte's polynomial. We also show that bounded portions of our interlace polynomial can be evaluated in polynomial time for graphs of bounded clique-width. Its full evaluation is necessarly exponential just because of the size of the result.

cs.LO↗

arXiv2006

The recognizability of sets of graphs is a robust property

Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of context-free sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) context-free graph grammars, to Vertex Replacement (VR) context-free graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifier-free formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HR-recognizability and VR-recognizability coincide. The same combinatorial condition equates HR-context-free and VR-context-free sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures.

cs.LO↗