  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.

Combinatorics