A separator theorem for graphs of bounded genus
Journal of Algorithms
Definability with bounded number of bound variables
Information and Computation
SIAM Journal on Discrete Mathematics
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Treewidth and Pathwidth of Permutation Graphs
SIAM Journal on Discrete Mathematics
Isomorphism testing for embeddable graphs through definability
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Computational Complexity of Ehrenfeucht-Fraïssé Games on Finite Structures
Proceedings of the 12th International Workshop on Computer Science Logic
Fixed-Point Logics on Planar Graphs
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
How complex are random graphs in first order logic?
Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
Generating Outerplanar Graphs Uniformly at Random
Combinatorics, Probability and Computing
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
Parameterized Complexity
The first order definability of graphs: upper bounds for quantifier depth
Discrete Applied Mathematics
First-Order Definability of Trees and Sparse Random Graphs
Combinatorics, Probability and Computing
Decomposable graphs and definitions with no quantifier alternation
European Journal of Combinatorics
Testing graph isomorphism in parallel by playing a game
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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We say that a first order formula φ defines a graph G if φ is true on G and false on every grap G' non-isomorphic with G. Let D(G) be the minimum quantifier rank of a such formula. We prove that, if G is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G) = O (log n), where n denotes the order of G. This bound is optimal up to a constant factor. If h is a constant, for connected graphs with no minor Kh and degree O (√nlog n), we prove the bound D(G) = O (√n). This result applies to planar graphs and, more generally, to graphs of bounded genus.Our proof techniques are based on the characterization of the quantifier rank as the length of the Ehrenfeucht game on non-isomorphic graphs. We use the separator theorems to design a winning strategy for Spoiler in this game.