Given a fixed graph H, the H-Free Edge Deletion (resp., Completion, Editing) problems ask whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most k edges so that the resulting graph is H-free, i.e., contains no induced subgraph isomorphic to H. These H-free edge modification problems are well known to be FPT for every fixed H. In this paper, we study...

The problem we consider in this paper was raised in [3]. Suppose T is a torus on the boundary of an orientable 3-manifold X, and S is a surface on ∂X − T which is incompressible in X. A slope γ is the isotopy class of a nontrivial simple closed curve on T . Denote by X(γ) the manifold obtained by attaching a solid torus to X so that γ is the slope of the boundary of a meridian disc. Given two s...

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with nonempty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the existence of alternating knots with noninteger boundary slopes, and a proof of ...

1. Introduction. What makes one object more complex than another? Kolmogorov complexity, or program-size complexity, provides one of many possible answers to this fundamental question. In this theory, whose foundations have been developed independently by R. the complexity of an object is defined as the length of its shortest effective description, which is the minimum number of symbols that mu...