Complexity of the cluster deletion problem on some subclasses of chordal graphs

Flavia Bonomo (1, 2) , Guillermo Duran (1, 3, 4) , Mario Valencia-Pabon (5, 6)

1 CONICET - Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires]
2 Departamento de Computación [Buenos Aires]
3 Departamento de Matemática [Buenos Aires]
4 DII - Departamento de Ingenieria Industrial [Santiago]
5 LIPN - Laboratoire d'Informatique de Paris-Nord
6 ORPAILLEUR - Knowledge representation, reasonning

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractability (FPT), where it is commonly known as the CLUSTER DELETION problem. Many of the recently-developed FPT algorithms rely on being able to solve CLUSTER DELETION in polynomial-time on restricted graph structures. In this paper, the complexity of the CLUSTER DELETION problem is investigated for the family of chordal graphs. It is shown that this problem is NP-complete for edge-weighted split graphs, edge-weighted interval graphs and edge-unweighted chordal graphs. We also prove that the CLUSTER DELETION is an NP-complete problem for edge-weighted cographs. Some polynomial-time solvable cases are also identified, in particular CLUSTER DELETION for unweighted split graphs, unweighted proper interval graphs and weighted block graphs.

Mots clés: Cographs; Submodular functions; Chordal graphs; Split graphs; Block graphs; Interval graphs; NP-completeness; Edge-deletion; Cluster deletion; Cliques

Domaines: Informatique [cs]; Mathématiques [math]

Flavia Bonomo, Guillermo Duran, Mario Valencia-Pabon. Complexity of the cluster deletion problem on some subclasses of chordal graphs. Theoretical Computer Science, 2015, 600, pp.59-69. ⟨hal-01102512⟩