Minimum feedback vertex set and acyclic coloring

  • Authors:

  • Guillaume Fertin;Emmanuel Godard;André Raspaudz

  • Affiliations:

  • IRIN UPRES-EA 2157, Université de Nantes, 2 rue de la Houssini, BP 92208, F44322 Nantes Cedex 3, France;LaBRI U.M.R. 5800, Université Bordeaux 1, 351 Cours de la Libèration, F33405 Talence Cedex, France;LaBRI U.M.R. 5800, Université Bordeaux 1, 351 Cours de la Libèration, F33405 Talence Cedex, France

  • Venue:
  • Information Processing Letters
  • Year:
  • 2002

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Abstract

In the feedback vertex set problem, the aim is to minimize, in a connected graph G = (V, E), the cardinality of the set V(G) ⊆ V, whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2.