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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 70/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Grid-Like Minors We will therefore use grid-like minors instead of grids. Theorem. (Reed, Wood) Any graph G of tree-width ≥ k 5 contains two sets P,Q of disjoint paths such that their intersection graph I(P,Q) contains a K k -minor.
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 71/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Grid-Like Minors We will therefore use grid-like minors instead of grids. Theorem. (Reed, Wood) Any graph G of tree-width ≥ k 5 contains two sets P,Q of disjoint paths such that their intersection graph I(P,Q) contains a K k -minor. Theorem. (K., Tazari 10) There is a constant c and a polynomial-time algorithm which, given a graph G with tw(G) > c · k 12 , computes a (topological) grid-like minor of order k in G. If we allow randomised algorithms we can reduce the tree-width to tw(G) > c ′ · l 5 to either • find a model of K l in G or • a (top.) grid-like minor (P,Q) of order l.
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