Slides.

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