Efficient algorithms for decomposing graphs under degree constraints

  • Authors:

  • Cristina Bazgan;Zsolt Tuza;Daniel Vanderpooten

  • Affiliations:

  • LAMSADE, Université Paris-Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary and Department of Computer Science, University of Pannonia, Veszprém, Hungary;LAMSADE, Université Paris-Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d(v)=a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that d"A(v)=a(v) for every v@?A and d"B(v)=b(v) for every v@?B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d(v)=a+b (a,b=1) or just d(v)=a+b-1 (a,b=2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.