Building large k-cores from sparse graphs
Fedor V. Fomin, Danil Sagunov, Kirill Simonov · Journal of Computer and System Sciences · 2022
A k-core of a graph G is the maximal induced subgraph in which every vertex has degree at least k. In the Edge k-Core optimization problem, we are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices, by adding at most b edges. While Edge k-Core is known to be computationally hard in general, we show that there are efficient algorithms when the k-core has to be constructed from a sparse graph with some structural properties. Our results are as follows. When the input graph is a forest, Edge k-Core is solvable in polynomial time. Edge k-Core is fixed-parameter tractable (FPT) when parameterized by the minimum size of a vertex cover in the input graph. Edge k-Core is FPT when parameterized by the treewidth of the graph plus k.