Optimal node ranking of trees in linear time
Information Processing Letters
The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs
European Journal of Combinatorics
Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Duality theorems for finite structures (characterising gaps and good characterisations)
Journal of Combinatorial Theory Series B
Proceedings of the 12th International Workshop on Computer Science Logic
On the order of countable graphs
European Journal of Combinatorics
On preservation under homomorphisms and unions of conjunctive queries
PODS '04 Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
On preservation under homomorphisms and unions of conjunctive queries
Journal of the ACM (JACM)
Linear time low tree-width partitions and algorithmic consequences
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Tree-depth, subgraph coloring and homomorphism bounds
European Journal of Combinatorics
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Grad and classes with bounded expansion I. Decompositions
European Journal of Combinatorics
Grad and classes with bounded expansion II. Algorithmic aspects
European Journal of Combinatorics
Grad and classes with bounded expansion III. Restricted graph homomorphism dualities
European Journal of Combinatorics
On forbidden subdivision characterizations of graph classes
European Journal of Combinatorics
Homomorphism preservation theorems
Journal of the ACM (JACM)
Fraternal augmentations, arrangeability and linear Ramsey numbers
European Journal of Combinatorics
European Journal of Combinatorics
Preservation under extensions on well-behaved finite structures
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Finite model theory on tame classes of structures
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
European Journal of Combinatorics
Characterisations and examples of graph classes with bounded expansion
European Journal of Combinatorics
FPT algorithms for domination in biclique-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Enumerating with constant delay the answers to a query
Proceedings of the 16th International Conference on Database Theory
Enumeration of first-order queries on classes of structures with bounded expansion
Proceedings of the 32nd symposium on Principles of database systems
Testing first-order properties for subclasses of sparse graphs
Journal of the ACM (JACM)
Interpreting nowhere dense graph classes as a classical notion of model theory
European Journal of Combinatorics
Distance-two coloring of sparse graphs
European Journal of Combinatorics
| Hi-index | 0.00 |
In this paper, we define and analyze the nowhere dense classes of graphs. This notion is a common generalization of proper minor closed classes, classes of graphs with bounded degree, locally planar graphs, classes with bounded expansion, to name just a few classes which are studied extensively in combinatorial and computer science contexts. In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes. This classification is surprisingly stable as it can be expressed in terms of the most commonly used basic combinatorial parameters, such as the independence number @a, the clique number @w, and the chromatic number @g. The remarkable stability of this notion and its robustness has a number of applications to mathematical logic, complexity of algorithms, and combinatorics. We also express the nowhere dense versus somewhere dense dichotomy in terms of edge densities as a trichotomy theorem.