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Grad and classes with bounded expansion. II: Algorithmic aspects. (English) Zbl 1185.05131
Summary: Classes of graphs with bounded expansion have been introduced in [(1) J. Nešetřil and P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, The grad of a graph and classes with bounded expansion. A. Raspaud (ed.) et al., 7th international colloquium on graph theory, Hyeres, France, September 12–16, 2005. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 22, 101–106 (2005; Zbl 1182.05102), (2) 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 \(\mathcal C\) and any integer \(p\) there exists a constant \(N({\mathcal C},p)\) so that the vertex set of any graph \(G\in{\mathcal C}\) may be partitioned into at most \(N({\mathcal C},p)\) parts, any \(i\leq p\) parts of them induce a subgraph of tree-width at most \((i-1)\) [(2)] (actually, of tree-depth [J. Nešetřil and P. Ossona de Mendez, Eur. J. Comb. 27, No. 6, 1022–1041 (2006; Zbl 1089.05025)] at most \(i\), which is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [J. Nešetřil and 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 \(\mathcal C\) with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed \(\mathcal C\) 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 \(\phi\) be a first-order logic sentence. We prove that any fixed graph property of type
\[ \text{``}\exists X:(|X|\leq p)\wedge (G[X]\vDash\phi)\text{''} \]
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.

MSC:
05C83 Graph minors
05C75 Structural characterization of families of graphs
05C85 Graph algorithms (graph-theoretic aspects)
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