The concept of exploiting properties that a graph possesses locally can be taken even further. Dawar, Grohe and Kreutzer in Locally Excluding a Minor considered classes of graphs that locally exclude a minor and Dvorak, Kral and Thomas in Deciding first-order properties for sparse graphs considered classes of graphs that have (locally) bounded expansion.
Both of those classes are subsumed by classes of nowhere dense graphs, introduced by Nesetril and Ossona de Mendez.
Grohe announced this week at the Highlights conference that Grohe, Kreutzer and Siebertz. have proved that every property of graphs definable in first-order logic can be solved in almost linear time on nowhere dense classes of graphs. This implies many fixed-parameter tractability results on nowhere dense graphs, e.g. for the (connected) dominating set and digraph kernel (both parameterized by the size of the solution), Steiner tree (parameterized by the size of the tree) and circuit satisfiability (parameterized by the depth of the circuit and the Hamming weight of the solution).