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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 34/81<br />
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
First-Order Logic on Bounded Degree Graphs<br />
Theorem: (Seese, 1996)<br />
Let C be a class of graphs of maximum degree at most d ≥ 1.<br />
MC(FO, C)<br />
Input:<br />
Parameter:<br />
Problem:<br />
Graph G ∈ C, ϕ ∈ FO<br />
|ϕ|<br />
Decide G |= ϕ<br />
is fixed-parameter tractable (linear time fpt algorithm).<br />
Proof. By Gaifman’s theorem it suffices to consider formulae of the form<br />
∧<br />
k∧<br />
∃x 1 ...∃x m dist(x i , x j ) > 2r ∧ ψ(x i )<br />
for some r-local formula ψ(x).<br />
1≤i
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