Approximation algorithms for clique-transversal sets and clique-independent sets in cubic graphs

A clique-transversal set S of a graph G is a subset of vertices intersecting all the cliques of C. where a clique is a complete subgraph maximal under inclusion and having at least two vertices. A clique-independent set of the graph G is a set of pairwise disjoint cliques of G. The clique-transversal number tau(C)(G) of G is the cardinality of the smallest clique-transversal set in G and the clique-independence number alpha(C)(G) of G is the cardinality of the largest clique-independent set in G. This paper proves that determining tau(C)(C) and alpha(C)(G) is NP-complete for a cubic planar graph G of girth 3. Further we propose two approximation algorithms for determining tau(C)(G) and alpha(C)(G) in a cubic graph G. (C) 2011 Elsevier B.V. All rights reserved.