Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs.

Let C and D be hereditary graph classes. Consider the following problem: given a graph G ∈ D , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to 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:the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in D admit balanced separators of size governed by their density, e.g., O ( Δ ) or O ( m ) , where Δ 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 C and D :a largest induced forest in a P t -free graph can be found in 2 O ~ ( n 2 / 3 ) time, for every fixed t; anda largest induced planar graph in a string graph can be found in 2 O ~ ( n 2 / 3 ) time.