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

Julien Bensmail (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Given a graph G, a sequence tau = (n_1, ..., 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, ..., 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 >= 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 Gyori and Lovasz 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 Pi_2^p-complete.

Mots clés: arbitrarily partitionable graphs; partition into connected subgraphs; partition under vertex prescriptions; complexity \and polynomial hierarchy

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail. On the complexity of partitioning a graph into a few connected subgraphs. Journal of Combinatorial Optimization, 2014, A paraître, http://link.springer.com/journal/10878. ⟨hal-00762612v2⟩