On the complexity of partitioning a graph into a few connected subgraphs
Résumé
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, that is 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 study the computational complexity of some decision problems related to the previous definition. In particular, we show that deciding whether a graph can be partitioned into several connected subgraphs is an NP-complete problem even when a constant number c of parts with c >= 2 is requested, or a constant number of vertex-membership constraints must be satisfied. We additionally introduce a Pi_2^p-complete graph partition problem asking whether some partial realizations of tau in G can be extended to obtain whole realizations of tau in G.
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