Classes of graphs with bounded expansion generalize both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N( C,p) so that the vertex set of any graph G∈C may be partitioned into at most N(C,p) parts, any i≤ p parts of them induce a subgraph of tree-width at most (i-1) (actually, of tree-depth at most i, what is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities. We give here a simple algorithm to compute such partitions and prove that if we restrict the input graph to some fixed class C with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed C and p). This result is applied to get a linear time algorithm for the subgraph isomorphism problem with fixed pattern and input graphs in a fixed class with bounded expansion. More generally, let φ be a first order logic sentence. We prove that any fixed graph property of type ``∃ X: (|X|≤ p) /\ (G[X]⊧φ)'' may be decided in linear time for input graphs in a fixed class with bounded expansion.
