Odd Cuts in Bipartite Grafts II: Structure and Universality of Decapital Distance Components

This paper is the second in a series of papers characterizing the maximum packing of $T$-cuts in bipartite grafts, following the first paper (N.~Kita, ``Tight cuts in bipartite grafts~I: Capital distance components,‘’ {arXiv:2202.00192v2}, 2022). Given a graft $(G,T)$, a minimum join $F$, and a specified vertex $r$ called the root, the distance components of $(G,T)$ are defined as subgraphs of $G$ determined by the distances induced by $F$. A distance component is called {\em capital} if it contains the root; otherwise, it is called {\em decapital}. In our first paper, we investigated the canonical structure of capital distance components in bipartite grafts, which can be described using the graft analogue of the Kotzig–Lovász decomposition. In this paper, we provide the counterpart structure for the decapital distance components. We also establish a necessary and sufficient condition for two vertices $r$ and $r^{\mathrm{\prime }}$ under which a decapital distance component with respect to root $r$ is also a decapital distance component with respect to root $r^{\mathrm{\prime }}$. As a consequence, we obtain that the total number of decapital distance components in a bipartite graft, taken over all choices of root, is equal to twice the number of edges in a minimum join of the graft.