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Partitioning Harary graphs into connected subgraphs containing prescribed vertices

Abstract : A graph G is arbitrarily partitionable (AP for short) if for every partition (n_1, n_2, ..., n_p) of |V(G)| there exists a partition (V_1, V_2, ..., V_p) of V(G) such that each V_i induces a connected subgraph of G with order n_i. If, additionally, k of these subgraphs (k <= p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). Every AP+k graph on n vertices is (k+1)-connected, and thus has at least ceil(n(k+1)/2) edges. We show that there exist AP+k graphs on n vertices and ceil(n(k+1)/2) edges for every k >= 1 and n >= k.
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Olivier Baudon, Julien Bensmail, Eric Sopena. Partitioning Harary graphs into connected subgraphs containing prescribed vertices. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, 16 (3), pp.263-278. ⟨hal-00687607v7⟩

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