Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Let $${\mathcal {C}}$$C and $${\mathcal {D}}$$D be 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 $${\mathcal {C}}$$C are sparse, i.e., they have linearly many edges in terms of the number of vertices; the graphs in $${\mathcal {D}}$$D admit balanced separators of size governed by their density, e.g., $${\mathcal {O}}(\varDelta )$$O(Δ) or $${\mathcal {O}}(\sqrt{m})$$O(m) , where $$\varDelta$$Δ andmdenote 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 $${\mathcal {C}}$$C and $${\mathcal {D}}$$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; and a largest induced planar graph in a string graph can be found in $$2^{\tilde{{\mathcal {O}}}(n^{2/3})}$$2O~(n2/3) time.