Induced subgraphs of bounded treewidth and the container method
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A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By P_t we denote a path on t vertices. In this paper we give polynomial-time algorithms for the following problems: the Maximum Weight Independent Set problem in long-hole-free graphs, and the Feedback Vertex Set problem in P_5-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended C_5 is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let C be the class of graphs excluding an extended C_5 and holes of length at least 6 as induced subgraphs; C contains all long-hole-free graphs and all P_5-free graphs. We show that, given an n-vertex graph G ∈C with vertex weights and an integer k, one can in time n^(k) find a maximum-weight induced subgraph of G of treewidth less than k. This implies both aforementioned results.
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