Handbook of combinatorics (vol. 2)
Graph classes: a survey
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
The first order definability of graphs with separators via the Ehrenfeucht game
Theoretical Computer Science - Game theory meets theoretical computer science
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
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
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Let D(G) be the minimum quantifier depth of a first order sentence @F that defines a graph G up to isomorphism. Let D"0(G) be the version of D(G) where we do not allow quantifier alternations in @F. Define q"0(n) to be the minimum of D"0(G) over all graphs G of order n. We prove that for all n we have log^*n-log^*log^*n-2@?q"0(n)@?log^*n+22, where log^*n is equal to the minimum number of iterations of the binary logarithm needed to bring n to 1 or below. The upper bound is obtained by constructing special graphs with modular decomposition of very small depth.