On maximum planar induced subgraphs

  • Authors:

  • Luerbio Faria;Celina M. Herrera de Figueiredo;Sylvain Gravier;Candido F. X. de Mendonça;Jorge Stolfi

  • Affiliations:

  • Departamento de Matemática, Faculdade de Formação de Professores, Universidade do Estado do Rio de Janeiro, Brazil;Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Brazil;CNRS, GeoD research group, "Maths à modeler" project, Laboratoire Leibniz, France;Departamento de Informática, Universidade Estadual de Maringá, Brazil;Instituto de Computação, Universidade Estadual de Campinas, Brazil

  • Venue:
  • Discrete Applied Mathematics - Special issue: Traces of the Latin American conference on combinatorics, graphs and applications: a selection of papers from LACGA 2004, Santiago, Chile
  • Year:
  • 2006

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Abstract

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G'. In this case G' is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)≤ k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)≤ k. We prove that unless P = NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time 3/4-approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.