We study algorithmic properties of the graph class CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document}, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document}. More precisely, we identify a large class of optimization problems on CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document} solvable in time 2O(klogk)·nO(1)\documentclass[12pt]{minimal}
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\begin{document}$$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$\end{document}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document} when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document} graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document} graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document} graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of CHORDAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Chordal}}{-ke}$$\end{document}, namely, INTERVAL-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Interval}}{-ke}$$\end{document} and SPLIT-ke\documentclass[12pt]{minimal}
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\begin{document}$${\textsc {Split}}{-ke}$$\end{document} graphs.
