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Splitting Number is NP-Complete

  • L. Faria
  • C. M. H. de Figueiredo
  • C. F. X. Mendonça
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k ≥ 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v either to v1 or to v2. We prove that the splitting number decision problem is NP-complete, even when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs also implies NP-completeness for graphs not containing a subdivision of K5 as a subgraph.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • L. Faria
    • 1
  • C. M. H. de Figueiredo
    • 2
  • C. F. X. Mendonça
    • 3
  1. 1.Faculdade de Formação de Professores/UERJ and COPPE/UFRJBrazil
  2. 2.Instituto de Matemática and COPPEUniversidade Federal do Rio de JaneiroBrazil
  3. 3.Instituto de ComputaçãoUniversidade Estadual de CampinasBrazil

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