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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 20/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Feferman-Vaught Style Theorems Theorem. Let G, H be graphs, v ∈ V(G) u ∈ V(G) such that u = V(G)∩V(H) w ∈ V(H) For all q ≥ 0, tp q (G∪H, uvw) is determined by tp q (G, uv) and tp q (uw) Furthermore, there is an algorithm that computes tp q (G∪H, uvw) from tp q (G, uv) and tp q (uw). This suggests a model-checking algorithm on graphs which can recursively be decomposed into sub-graphs of constant size. graphs of bounded tree-width
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 21/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Tree-Width The tree-width of a graph measures its similarity to a tree. A graph has tree-width ≤ k if it can be covered by sub-graphs of size ≤ (k + 1) in a tree-like fashion.
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