TY - JOUR

T1 - Vertex set partitions preserving conservativeness

AU - Ageev, A. A.

AU - Kostochka, A. V.

N1 - Funding Information:
2This author finished his part of the work during a stay at SFB343 ‘‘Diskrete Strukturen in der Mathematik’’ of Bielefeld University. His work was also partially supported by Grant 96-01-01614 under the Russian Foundation for Basic Research.

PY - 2000/11

Y1 - 2000/11

N2 - 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 Theory34, 357-364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183-191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 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).

AB - 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 Theory34, 357-364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183-191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 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).

KW - Undirected graph; T-join; T-cut; conservative weighting

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U2 - 10.1006/jctb.2000.1972

DO - 10.1006/jctb.2000.1972

M3 - Article

AN - SCOPUS:0034311708

VL - 80

SP - 202

EP - 217

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -