# The Complexity of Cluster Vertex Splitting and Company

Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex v means to remove it and to add two new copies of v and to make each previous neighbor of v adjacent with at least one of the copies. In this work, we study the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem, which also has been independently studied in different contexts,is NP-complete. Based on a previous so-called critical-clique lemma, we leverage our hardness result to show that Cluster Editing with Vertex Splitting is also NP-complete, resolving an open question by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the critical-clique lemma is flawed and we give a counterexample. Our hardness result also holds under a version of the critical-clique lemma to which we currently do not have a counterexample. On the positive side, we show that Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to the number of splits.

• 1 publication
• 7 publications
• 1 publication
• 1 publication
• 28 publications
• 10 publications
• 1 publication
research
01/01/2019

### On the Parameterized Cluster Editing with Vertex Splitting Problem

In the Cluster Editing problem, a given graph is to be transformed into ...
research
02/24/2022

### Planarizing Graphs and their Drawings by Vertex Splitting

The splitting number of a graph G=(V,E) is the minimum number of vertex ...
research
10/11/2020

### On Structural Parameterizations of Load Coloring

Given a graph G and a positive integer k, the 2-Load coloring problem is...
research
10/04/2017

### The Parameterized Complexity of Centrality Improvement in Networks

The centrality of a vertex v in a network intuitively captures how impor...
research
12/01/2017

### Graph Homomorphism Reconfiguration and Frozen H-Colourings

For a fixed graph H, the reconfiguration problem for H-colourings (i.e. ...
research
02/28/2023

### Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

Motivated by the planarization of 2-layered straight-line drawings, we c...
research
09/12/2020

### Open Problem: Average-Case Hardness of Hypergraphic Planted Clique Detection

We note the significance of hypergraphic planted clique (HPC) detection ...