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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 45/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Localisation of Graph Invariants Definition: Let f : GRAPHS → N be a graph invariant. We define its localisation loc f : GRAPHS×N → N as { ( loc f (G, r) := max f G [ N r (v) ]) } : v ∈ V(G) . A class C of graphs has bounded local f , if there is a computable function h : N → N such that loc f (G, r) ≤ h(r) for all G ∈ C and r ∈ N. Example: f : G ↦→ tw(G) tree-width of graphs { ( loc f (G, r) := max tw G [ N r (v) ]) } : v ∈ V(G) Bounded local tree-width
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 46/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Bounded Local Tree-Width Bounded local tree-width: f : G ↦→ tw(G) tree-width of graphs { ( loc f (G, r) := max tw G [ N r (v) ]) } : v ∈ V(G) Example: Every class of graphs of bounded degree has bounded local tree-width. Example: The class of planar graphs has bounded local tree-width. Theorem: Every planar graph of diameter r has tree-width at most 3r. (Baker)
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