2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 2530
author:  Oleg Pikhurko, Joel Spencer and Oleg Verbitsky 

title:  Decomposable graphs and definitions with no quantifier alternation 
keywords:  descriptive complexity of graphs, first order logic, Ehrenfeucht game on graphs, graph decompositions 
abstract: 
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
be a variant of
0
(G)
D(G)
where we do not allow quantifier alternations in
Φ
. Using large graphs decomposable in
complementconnected components by a short sequence of
serial and parallel decompositions, we show examples of
G
on
n
vertices with
D
. On the other hand, we prove a lower bound
0
(G)≤2
log
*
n+O(1)
D
for all
0
(G)≥
log
*
n
log
*
log
*
nO(1)
G
. Here
log
*
n
n
below 1.

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reference:  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. 2530 
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