Mathematical analysis

Dominating set

Parameterized complexity

Line graph

Upper and lower bounds

Treewidth

Pathwidth

Combinatorics

Physics

Graph

Theory of computation

Mathematics

Acoustics

Vertex cover

Degree (music)

Algorithm

Vertex (graph theory)

Discrete mathematics

Bounded function

Kernelization

Feedback vertex set

Algorithmica

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.

On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem