Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join
Nanano Kita
TL;DR
The paper provides a constructive characterization of grafts that admit a connected minimum join and an efficient algorithm to recognize them. It builds on distance-theoretic tools for grafts, notably Sebő's distance decomposition, and introduces the notion of rakes to capture the core structural components. The authors present a gluing-sum framework that combines a base graft with disjoint rakes to characterize all grafts with connected minimum joins, and they deliver a polynomial-time recognition algorithm that uses minimum $T$-joins and advanced distance computations to either output a connected minimum join or certify nonexistence. The results improve known time bounds and offer a practical route to compute connected minimum joins, with implications for Seymour's min-max formula and related packing problems in grafts.
Abstract
Minimum joins in a graft $(G, T)$, also known as minimum $T$-joins of a graph $G$, are said to be connected if they determine a connected subgraph of $G$. Grafts with a connected minimum join have gained interest ever since Middendorf and Pfeiffer showed that they satisfy Seymour's min-max formula for joins and $T$-cut packings; that is, in such grafts, the size of a minimum join is equal to the size of a maximum packing of $T$-cuts. In this paper, we provide a constructive characterization of grafts with a connected minimum join. We also obtain a polynomial time algorithm that decides whether a given graft has a connected minimum join and, if so, outputs one. Our algorithm has two bottlenecks; one is the time required to compute a minimum join of a graft, and the other is the time required to solve the single-source all-sink shortest path problem in a graph with conservative $\pm 1$-valued edge weights. Thus, our algorithm runs in $O(n(m + n\log n) )$ time. In the nondense case, it improves upon the time bound for this problem due to Sebő and Tannier that was introduced as an application of their results on metrics on graphs.