Grad and classes with bounded expansion II. Algorithmic aspects

Abstract

Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, O. Delmas (Eds.), 7th International Colloquium on Graph Theory, in: Electronic Notes in Discrete Mathematics, vol. 22, Elsevier (2005), pp. 101–106; J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)]. They generalize classes with forbidden topological minors (i.e. classes of graphs having no subgraph isomorphic to the subdivision of some graph in a forbidden family), and hence 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) [J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)] (actually, of tree-depth [J. Nešetřil, P. Ossona de Mendez, Tree depth, subgraph coloring and homomorphism bounds, European Journal of Combinatorics 27 (6) (2006) 1022–1041] at most i, which is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion III. Restricted dualities, European Journal of Combinatorics (2005) (submitted for publication)]. We give here a simple algorithm for computing 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 may be decided in linear time for input graphs in a fixed class with bounded expansion. 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. Also, extending several earlier results, we show that a class of graphs has sublinear separators if it has sub-exponential expansion. This result is best possible in general.