On a min–max theorem on bipartite graphs

AbstractFrank et al. (Math. Programming Stud. 22 (1984) 99–112) proved that for any connected bipartite graft (G,T), the minimum size of a T-join is equal to the maximum value of a partition of A, where A is one of the two colour classes of G. Their proof consists of constructing a partition of A of value |F|, by using a minimum T-join F. That proof depends heavily on the properties of distances in graphs with conservative weightings. We follow the dual approach, that is starting from a partition of A of maximum value k, we construct a T-join of size k. Our proof relies only on Tutte's theorem on perfect matchings. It is known (J. Combin. Theory Ser. B 61 (2) (1994) 263–271) that the results of Lovász on 2-packing of T-cuts, of Seymour on packing of T-cuts in bipartite graphs and in grafts that cannot be T-contracted onto (K4,V(K4)), and of Sebő on packing of T-borders are implied by this theorem of Frank et al. The main contribution of the present paper is that all of these results can be derived from Tutte's theorem

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and P=[X 1,..., X n] be a partition of V(G). Denote by G P the graph which has vertex set [X 1,..., X n], edge set E, and is obtained from G by identifying vertices in each class X i 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 proof

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and P = fX 1 ; : : : ; X n g be a partition of V (G). Denote by G/P the graph which has vertex set fX 1 ; : : : ; X n g, edge set E, and is obtained from G by identifying vertices in each class X i 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 obtained results are then used in new unified short proofs for a co-NP characterization of Seymour graphs [1], a theorem of Korach and Penn [5], a theorem of Korach [4], and a theorem of Kostochka [6]