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INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
Localisation of Graph Invariants<br />
Let f : GRAPHS → N be a induced subgraph monotone graph invariant.<br />
Theorem: Let C be a class of graphs such that the following is fpt:<br />
MC(FO, f )<br />
Input:<br />
Parameter:<br />
Problem:<br />
ϕ ∈ FO, Graph G ∈ C<br />
|ϕ|+f(G)<br />
Decide G |= ϕ<br />
Then first-order model checking is fixed-parameter tractable on C.<br />
Follows immediately from the following theorem proved before.<br />
Theorem: Let C be a class of graphs such that the following is fpt:<br />
LOCAL-FO-MC<br />
Input: ϕ ∈ FO, Graph G ∈ C,v 1 ,...,v k ∈ V(G), and r ∈ N<br />
Parameter: r + k +|ϕ|<br />
Problem: Decide G [ Nr G (v 1 ,...,v k )] |= ϕ<br />
Then first-order model checking is fixed-parameter tractable on C.<br />
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 47/81
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