Abstract
A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by τ C (G) and α C (G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τ C (H)=α C (H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τ C (G) and α C (G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τ C (G)−α C (G) is arbitrarily large.
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Durán, G., Lin, M.C. & Szwarcfiter, J.L. On Clique-Transversals and Clique-Independent Sets. Annals of Operations Research 116, 71–77 (2002). https://doi.org/10.1023/A:1021363810398
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DOI: https://doi.org/10.1023/A:1021363810398