Building large k-cores from sparse graphs

by   Fedor V. Fomin, et al.

A popular model to measure network stability is the k-core, that is the maximal induced subgraph in which every vertex has degree at least k. For example, k-cores are commonly used to model the unraveling phenomena in social networks. In this model, users having less than k connections within the network leave it, so the remaining users form exactly the k-core. In this paper we study the question whether it is possible to make the network more robust by spending only a limited amount of resources on new connections. A mathematical model for the k-core construction problem is the following 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. The previous studies on Edge k-Core demonstrate that the problem is computationally challenging. In particular, it is NP-hard when k=3, W[1]-hard being parameterized by k+b+p (Chitnis and Talmon, 2018), and APX-hard (Zhou et al, 2019). Nevertheless, we show that there are efficient algorithms with provable guarantee when the k-core has to be constructed from a sparse graph with some additional structural properties. Our results are 1) When the input graph is a forest, Edge k-Core is solvable in polynomial time; 2) Edge k-Core is fixed-parameter tractable (FPT) being parameterized by the minimum size of a vertex cover in the input graph. On the other hand, with such parameterization, the problem does not admit a polynomial kernel subject to a widely-believed assumption from complexity theory; 3) Edge k-Core is FPT parameterized by tw+k. This improves upon the result of Chitnis and Talmon by not requiring b to be small. Each of our algorithms is built upon a new graph-theoretical result interesting in its own.


page 1

page 2

page 3

page 4


A Parameterized Complexity View on Collapsing k-Cores

We study the NP-hard graph problem Collapsed k-Core where, given an undi...

K-Core Maximization through Edge Additions

A popular model to measure the stability of a network is k-core - the ma...

K-Core Minimization: A Game Theoretic Approach

K-cores are maximal induced subgraphs where all vertices have degree at ...

On the Parameterized Complexity of Computing st-Orientations with Few Transitive Edges

Orienting the edges of an undirected graph such that the resulting digra...

Reducing the Vertex Cover Number via Edge Contractions

The CONTRACTION(vc) problem takes as input a graph G on n vertices and t...

Parameter-free Structural Diversity Search

The problem of structural diversity search is to nd the top-k vertices w...

Hierarchical team structure and multidimensional localization (or siloing) on networks

Knowledge silos emerge when structural properties of organizational inte...

Please sign up or login with your details

Forgot password? Click here to reset