Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Let [Formula: see text] and [Formula: see text] be hereditary graph classes. Consider the following problem: given a graph [Formula: see text] , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to [Formula: see text] . We prove that it can be solved in [Formula: see text] time, where n is the number of vertices of G, if the following conditions are satisfied: the graphs in [Formula: see text] are sparse, i.e., they have linearly many edges in terms of the number of vertices; the graphs in [Formula: see text] admit balanced separators of size governed by their density, e.g., [Formula: see text] or [Formula: see text] , where [Formula: see text] and m denote the maximum degree and the number of edges, respectively; and the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes [Formula: see text] and [Formula: see text] : a largest induced forest in a [Formula: see text] -free graph can be found in [Formula: see text] time, for every fixed t; and a largest induced planar graph in a string graph can be found in [Formula: see text] time.