Bivariate complexity analysis of Almost Forest Deletion

Abstract In this paper we study a generalization of classic Feedback Vertex Set problem in the realm of multivariate complexity analysis. We say that a graph F is an l-forest if we can delete at most l edges from F to get a forest. That is, F is at most l edges away from being a forest. In this paper we introduce the Almost Forest Deletion problem, where given a graph G and integers k and l, the question is whether there exists a subset of at most k vertices such that its deletion leaves us an l-forest. We show that this problem admits an algorithm with running time 2 O ( k + l ) n O ( 1 ) and a kernel of size O ( k l ( k + l ) ) . We also show that the problem admits a 2 O ( tw ) n O ( 1 ) algorithm on bounded treewidth graphs, using which we design a subexponential algorithm for the problem on planar graphs.

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