Summary: Let \({\mathcal{C}}\) and \({\mathcal{D}}\) be hereditary graph classes. Consider the following problem: given a graph \(G\in{\mathcal{D}} \), find a largest, in terms of the number of vertices, induced subgraph of \(G\) that belongs to \({\mathcal{C}} \). We prove that it can be solved in \(2^{o(n)}\) time, where \(n\) is the number of vertices of \(G\), if the following conditions are satisfied:
i) the graphs in \({\mathcal{C}}\) are sparse, i.e., they have linearly many edges in terms of the number of vertices;
ii) the graphs in \({\mathcal{D}}\) admit balanced separators of size governed by their density, e.g., \({\mathcal{O}}(\varDelta )\) or \({\mathcal{O}}(\sqrt{m})\), where \(\varDelta\) and \(m\) denote the maximum degree and the number of edges, respectively and
iii) 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 \({\mathcal{C}}\) and \({\mathcal{D}} \):

a) a largest induced forest in a \(P_t\)-free graph can be found in \(2^{\tilde{{\mathcal{O}}}(n^{2/3})}\) time, for every fixed \(t\); and
b) a largest induced planar graph in a string graph can be found in \(2^{\tilde{{\mathcal{O}}}(n^{2/3})}\) time.