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Vertex set partitions preserving conservativeness. (English) Zbl 1027.05078
Summary: Let \(G\) be an undirected graph and \({\mathcal P}= \{X_1,\dots, X_n\}\) be a partition of \(V(G)\). Denote by \(G/{\mathcal P}\) the graph which has vertex set \(\{X_1,\dots, X_n\}\), edge set \(E\), and is obtained from \(G\) by identifying vertices in each class \(X_i\) of the partition \({\mathcal P}\). Given a conservative graph \((G,{\mathbf w})\), we study vertex set partitions preserving conservativeness, i.e., those for which \((G/{\mathcal P},{\mathbf w})\) is also a conservative graph. We characterize the conservative graphs \((G/{\mathcal P},{\mathbf w})\), where \({\mathcal P}\) 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 of A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997), a theorem of E. Korach (1994), a theorem of E. Korach and M. Penn (1992), and a theorem of A. V. Kostochka (1994).

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C75 Structural characterization of families of graphs
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