AbstractLet$${\mathcal {C}}$$Cand$${\mathcal {D}}$$Dbe hereditary graph classes. Consider the following problem: given a graph$$G\in {\mathcal {D}}$$G∈D, find a largest, in terms of the number of vertices, induced subgraph ofGthat belongs to$${\mathcal {C}}$$C. We prove that it can be solved in$$2^{o(n)}$$2o(n)time, wherenis the number of vertices ofG, if the following conditions are satisfied:the graphs in$${\mathcal {C}}$$Care sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in$${\mathcal {D}}$$Dadmit balanced separators of size governed by their density, e.g.,$${\mathcal {O}}(\varDelta )$$O(Δ)or$${\mathcal {O}}(\sqrt{m})$$O(m), where$$\varDelta$$Δandmdenote the maximum degree and the number of edges, respectively; andthe 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$${\mathcal {C}}$$Cand$${\mathcal {D}}$$D:a largest induced forest in a$$P_t$$Pt-free graph can be found in$$2^{\tilde{{\mathcal {O}}}(n^{2/3})}$$2O~(n2/3)time, for every fixedt; anda largest induced planar graph in a string graph can be found in$$2^{\tilde{{\mathcal {O}}}(n^{2/3})}$$2O~(n2/3)time.