Linear time low tree-width partitions and algorithmic consequences

Classes of graphs with bounded expansion have been introduced in [15], [12]. They 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) [12] (actually, of tree-depth [16] at most i, what is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [14].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.

Authors

• Jaroslav Nesetril
• Patrice Ossona de Mendez

Keywords

• graph minor
• subgraph isomorphism
• coloration
• bounded expansion
• first order logic
• tree-width
• fraternal augmentation