Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Abstract

Let $C$ and $D$ be hereditary graph classes. Consider the following problem: given a graph $G\in D$ , 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 $2^{o\left(n\right)}$ 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., $O\left(\Delta \right)$ or $O\left(\sqrt{m}\right)$ , where $\Delta$ 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 $2^{\tilde{O}\left(n^{2/3}\right)}$ time, for every fixed t; anda largest induced planar graph in a string graph can be found in $2^{\tilde{O}\left(n^{2/3}\right)}$ time.

Keywords: P t -free graphs; Feedback vertex set; String graphs; Subexponential algorithm.