Tight Cuts in Bipartite Grafts I: Capital Distance Components

Abstract

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 {\em distance components}. Seb\"o 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 {\em 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-Lov\'asz canonical decomposition.