Odd Cuts in Bipartite Grafts II: Structure and Universality of Decapital Distance Components
Nanao Kita
TL;DR
This work develops a complete structural theory for decapital distance components in bipartite grafts, tying them to the Kotzig–Lovász decomposition and Sebő's distance framework. Using minimum joins $F$, $F$-beams, and a negative-set calculus, it establishes a canonical internal structure for decapital components, proves a root-shift universality, and derives a precise counting identity that the total number of decapital components across all roots equals $2ν(G,T)$. It also introduces root/antiroot classes and root fragments to organize these components and provides necessary-and-sufficient conditions for when a decapital component with respect to one root remains decapital with respect to another. Collectively, the results deepen the understanding of $T$-cut packing in bipartite grafts and supply tools for analyzing distance-induced component structure and minimum-join properties.
Abstract
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 $r'$ under which a decapital distance component with respect to root $r$ is also a decapital distance component with respect to root $r'$. 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.