Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and P=[X1 , ..., Xn] be a partition of V(G). Denote by G P the graph which has vertex set [X1 , ..., Xn], edge set E, and is obtained from G by identifying vertices in each class Xi of the partition P. Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G P, w) is also a conservative graph. We characterize the conservative graphs (G P, w), where P is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory 34, 357 364), a theorem of E. Korach and M. Penn (1992, Math. Programming 55, 183 191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B 62, 1 10), and a theorem of A. V. Kostochka (1994, in ``Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109 123, Kluwer Academic, Dordrecht).