Grad and Classes with Bounded Expansion II. Algorithmic Aspects
  Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, &#8711;r(G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general. 
Jaroslav Nesetril (KAM), Patrice Ossona De Mendez (CAMS)