Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Let $$ and $$ be hereditary graph classes. Consider the following problem: given a graph $$ , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to $$ . We prove that it can be solved in $$ time, where n is the number of vertices of G, if the following conditions are satisfied:the graphs in $$ are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in $$ admit balanced separators of size governed by their density, e.g., $$ or $$ , where $$ and m denote 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 $$ and $$ :a largest induced forest in a $$ -free graph can be found in $$ time, for every fixed t; anda largest induced planar graph in a string graph can be found in $$ time.