Tight Cuts in Bipartite Grafts I: Capital Distance Components
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This paper is the first from a series of papers that provide a characterization of maximum packings of T-cuts in bipartite graphs. Given a connected graph, a set T of an even number of vertices, and a minimum T-join, an edge weighting can be defined, from which distances between vertices can be defined. Furthermore, given a specified vertex called root, vertices can be classified according to their distances from the root, and this classification of vertices can be used to define a family of subgraphs called distance components. Sebö provided a theorem that revealed a relationship between distance components, minimum T-joins, and T-cuts. In this paper, we further investigate the structure of distance components in bipartite graphs. Particularly, we focus on capital distance components, that is, those that include the root. We reveal the structure of capital distance components in terms of the T-join analogue of the general Kotzig-Lovász canonical decomposition.
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