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On the complexity of partitioning a graph into a few connected subgraphs


Given a graph \(G\), a sequence \(\tau = (n_1, \ldots , n_p)\) of positive integers summing up to \(|V(G)|\) is said to be realizable in \(G\) if there exists a realization of \(\tau \) in \(G\), i.e. a partition \((V_1, \ldots , V_p)\) of \(V(G)\) such that each \(V_i\) induces a connected subgraph of \(G\) on \(n_i\) vertices. We first give a reduction showing that the problem of deciding whether a sequence with \(c\) elements is realizable in a graph is NP-complete for every fixed \(c \ge 2\). Thanks to slight modifications of this reduction, we then prove additional hardness results on decision problems derived from the previous one. In particular, we show that the previous problem remains NP-complete when a constant number of vertex-membership constraints must be satisfied. We then prove the tightness of an easiness result proved independently by Györi and Lovász regarding a similar problem. We finally show that another graph partition problem, asking whether several partial realizations of \(\tau \) in \(G\) can be extended to obtain whole realizations of \(\tau \) in \(G\), is \(\varPi _2^p\)-complete.

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  • Barth D, Baudon O, Puech J (2002) Decomposable trees: a polynomial algorithm for tripodes. Discret Appl Math 119(3):205–216

    Article  MATH  MathSciNet  Google Scholar 

  • Barth D, Fournier H (2006) A degree bound on decomposable trees. Discret Math 306(5):469–477

    Article  MATH  MathSciNet  Google Scholar 

  • Baudon O, Bensmail J, Przybyło J, Woźniak M (2012) Partitioning powers of traceable or Hamiltonian graphs (Preprint).

  • Dyer ME, Frieze AM (1985) On the complexity of partitioning graphs into connected subgraphs. Discret Appl Math 10:139–153

    Article  MATH  MathSciNet  Google Scholar 

  • Guttmann-Beck N, Hassin R (1997) Approximation algorithms for min–max tree partition. J Algorithms 24(2):266–286

    Article  MATH  MathSciNet  Google Scholar 

  • Györi E (1978) On division of graphs to connected subgraphs. In: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), vol I, pp 485–494, Colloq Math Soc János Bolyai, 18, North-Holland, Amsterdam.

  • Lovász L (1977) A homology theory for spanning trees of a graph. Acta Math Acad Sci Hung 30(3–4): 241–251

    Google Scholar 

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Correspondence to Julien Bensmail.

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Bensmail, J. On the complexity of partitioning a graph into a few connected subgraphs. J Comb Optim 30, 174–187 (2015).

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  • Arbitrarily partitionable graphs
  • Partition into connected subgraphs
  • Partition under vertex prescriptions
  • Complexity
  • Polynomial hierarchy