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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.
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 23/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Examples Example 1: Trees/Forests have tree-width 1 Proposition: Acyclic graphs are precisely the graphs of tree-width 1.
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