Subexponential-time algorithms for finding large induced sparse subgraphs

LetCandDbe hereditary graph classes. Consider the following problem: given a graphG∈ D,find a largest, in terms of the number of vertices, induced subgraph ofGthat belongs toC. Weprove that it can be solved in2o(n)time, wherenis the number of vertices ofG, if the followingconditions are satisfied:the graphs inCare sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs inDadmit balanced separators of size governed by their density, e.g.,O(∆)orO(√m), where∆andmdenote the maximum degree and the number of edges, respectively; andthe considered problem admits a single-exponential fixed-parameter algorithm when parameter-ized by the treewidth of the input graph.This leads, for example, to the following corollaries for specific classesCandD:a largest induced forest in aPt-free graph can be found in2 ̃O(n2/3)time, for every fixedt; anda largest induced planar graph in a string graph can be found in2 ̃O(n3/4) time.