Building large k-cores from sparse graphs.

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. [ABSTRACT FROM AUTHOR]