On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs

  • Authors:

  • L. Alcón;L. Faria;C. M. H. de Figueiredo;M. Gutierrez

  • Affiliations:

  • Departamento de Matemática, Universidad Nacional de La Plata, CC 172, (1900) La Plata, Argentina;Departamento de Matemática, Faculdade de Formação de Professores, Universidade do Estado do Rio de Janeiro, Brazil;COPPE/Sistemas, Universidade Federal do Rio de Janeiro, Brazil;CONICET, Argentina and Departamento de Matemática, Universidad Nacional de La Plata, CC 172, (1900) La Plata, Argentina

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

Clique-Helly and hereditary clique-Helly graphs are polynomial-time recognizable. Recently, we presented a proof that the clique graph recognition problem is NP-complete [L. Alcon, L. Faria, C.M.H. de Figueiredo, M. Gutierrez, Clique graph recognition is NP-complete, in: Proc. WG 2006, in: Lecture Notes in Comput. Sci., vol. 4271, Springer, 2006, pp. 269-277]. In this work, we consider the decision problems: given a graph G=(V,E) and an integer k=0, we ask whether there exists a subset V^'@?V with |V^'|=k such that the induced subgraph G[V^'] of G is, variously, a clique, clique-Helly or hereditary clique-Helly graph. The first problem is clearly NP-complete, from the above reference; we prove that the other two decision problems mentioned are NP-complete, even for maximum degree 6 planar graphs. We consider the corresponding maximization problems of finding a maximum induced subgraph that is, respectively, clique, clique-Helly or hereditary clique-Helly. We show that these problems are Max SNP-hard, even for maximum degree 6 graphs. We show a general polynomial-time 1@D+1-approximation algorithm for these problems when restricted to graphs with fixed maximum degree @D. We generalize these results to other graph classes. We exhibit a polynomial 6-approximation algorithm to minimize the number of vertices to be removed in order to obtain a hereditary clique-Helly subgraph.