Partition into cliques for cubic graphs: Planar case, complexity and approximation

• Published in: Discrete Applied Mathematics Vol. 156; no. 12; pp. 2270 - 2278
• Main Authors: Cerioli, M.R., Faria, L., Ferreira, T.O., Martinhon, C.A.J., Protti, F., Reed, B.
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 28.06.2008; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation; Approximations and expansions; Coloring; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Cubic planar graphs; Exact sciences and technology; Information retrieval. Graph; Mathematical analysis; Mathematics; Max SNP-hard; NP-complete; Partition into cliques; Sciences and techniques of general use; Theoretical computing; Coloring; Max SNP-hard; NP-complete; Approximation; Partition into cliques; Cubic planar graphs; Partition; Computer theory; Polynomial approximation; Graph clique; Cubic graph; Planar graph; Approximation algorithm; Optimization; Complexity; Polynomial time; Integer; Maximum; Graph degree; Subgraph; Combinatorics
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2007.10.015

Given a graph G = ( V , E ) and a positive integer k , the partition into cliques ( pic) decision problem consists of deciding whether there exists a partition of V into k disjoint subsets V 1 , V 2 , … , V k such that the subgraph induced by each part V i is a complete subgraph (clique) of G . In this paper, we establish both the NP-completeness of pic for planar cubic graphs and the Max SNP-hardness of pic for cubic graphs. We present a deterministic polynomial time 5 4 -approximation algorithm for finding clique partitions in maximum degree three graphs.