We study algorithmic properties of the graph class \({\textsc {Chordal}}{-ke}\), 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 \({\textsc {Chordal}}{-ke}\). More precisely, we identify a large class of optimization problems on \({\textsc {Chordal}}{-ke}\) solvable in time \(2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}\). 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 \({\textsc {Chordal}}{-ke}\) when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of \({\textsc {Chordal}}{-ke}\) 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 \({\textsc {Chordal}}{-ke}\) 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 \({\textsc {Chordal}}{-ke}\) graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of \({\textsc {Chordal}}{-ke}\), namely, \({\textsc {Interval}}{-ke}\) and \({\textsc {Split}}{-ke}\) graphs.

我们研究图类\（{\ textsc {Chordal}} {-ke} \）的算法特性，即，可以通过添加最多k个边或等效地将图类转化为弦图。最多填充k个。看来，在（\（{\ textsc {Chordal}} {-ke} \）的图上，有k个基本的难以解决的优化问题通过k接纳次指数算法进行参数化。更准确地说，我们在\（{\ textsc {Chordal}} {-ke} \）上可以及时解决\（2 ^ {{{\ mathcal {O}}}（\ sqrt {k} \ log k）} \ cdot n ^ {{\\ mathcal {O}}（1）} \）。此类问题的示例包括：找到独立的最大权重集，找到反馈顶点集或最小权重的奇数周期横向或找到最大诱导平面子图的问题。另一方面，我们表明，对于某些基本的优化问题，例如找到最佳的图形着色或找到最大的集团，当用k进行参数化时，它们是\（{\ textsc {Chordal}} {-ke} \）的FPT，但是除非ETH失败，否则不允许在k算法中使用次指数。除了次指数时间算法之外，从核化的角度来看（参数k为\（{\ textsc {Chordal}} {-ke} \）图的类别似乎也很有吸引力。）。尽管有可能表明大多数优化问题的加权变量都不接受\（{\ textsc {Chordal}} {-ke} \）图上的k个内核中的多项式，但这并不排除存在Turing内核化和未加权图的内核化。特别是，我们在\（{\ textsc {Chordal}} {-ke} \）图上为加权派系构造了多项式Turing核。对于（非加权）独立集，我们在\（{\ textsc {Chordal}} {-ke} \）的两个有趣子类上设计多项式内核，分别是\（{\ textsc {Interval}} {-ke} \）和\ （{\ textsc {Split}} {-ke} \）图。