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Let $mathcal{C}$ and $mathcal{D}$ be hereditary graph classes. Consider the following problem: given a graph $Ginmathcal{D}$, find a largest, in terms of the number of vertices, induced subgraph of $G$ that belongs to $mathcal{C}$. 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 $mathcal{C}$ are sparse, i.e., they have linearly many edges in terms of the number of vertices; * the graphs in $mathcal{D}$ admit balanced separators of size governed by their density, e.g., $mathcal{O}(Delta)$ or $mathcal{O}(sqrt{m})$, where $Delta$ and $m$ denote 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 $mathcal{C}$ and $mathcal{D}$: * a largest induced forest in a $P_t$-free graph can be found in $2^{tilde{mathcal{O}}(n^{2/3})}$ time, for every fixed $t$; and * a largest induced planar graph in a string graph can be found in $2^{tilde{mathcal{O}}(n^{3/4})}$ time.
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatr
We give algorithms with running time $2^{O({sqrt{k}log{k}})} cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices, (2) a cycle o
Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on $n$ vertices and a positive integer parameter $k$, find if there exist $k$ edges (arcs) whose delet
In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,vin V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-
We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $mathsf{BPTIME}$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised