## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 25-30

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 25-30

author: Oleg Pikhurko, Joel Spencer and Oleg Verbitsky Decomposable graphs and definitions with no quantifier alternation descriptive complexity of graphs, first order logic, Ehrenfeucht game on graphs, graph decompositions Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism in terms of the adjacency and the equality relations. Let D 0 (G) be a variant of D(G) where we do not allow quantifier alternations in Φ . Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of G on n vertices with D 0 (G)≤2 log * n+O(1) . On the other hand, we prove a lower bound D 0 (G)≥ log * n- log * log * n-O(1) for all G . Here log * n is equal to the minimum number of iterations of the binary logarithm needed to bring n below 1. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Oleg Pikhurko and Joel Spencer and Oleg Verbitsky (2005), Decomposable graphs and definitions with no quantifier alternation, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 25-30 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0106.ps.gz (72 K) dmAE0106.ps (174 K) dmAE0106.pdf (172 K)

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