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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 22/81<br />
INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS<br />
Tree-Width<br />
Definition:<br />
A tree-decomposition of a graph G is a pair T := (T,(B t ) t∈V T) where<br />
• T is a (directed) tree<br />
• B t ⊆ V(G) for all t ∈ V T<br />
such that<br />
1. for every edge {u, v} ∈ E(G) there is t ∈ V(T) with u, v ∈ B t<br />
2. for all v ∈ V(G) the set {t : v ∈ B t } is non-empty and connected.<br />
The width of T is max{|B t |−1 : t ∈ V(T)}<br />
The tree-width tw(G) of G is the minimal width of any of its tree-dec.<br />
Definition: A class C has bounded tree-width if there is a constant k ∈ N<br />
such that tw(G) ≤ k for all G ∈ C.