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 Seb\H{o} and Tannier that was introduced as an application of their results on metrics on graphs.

翻译：在嫁接图$(G, T)$中，最小连接（亦称图$G$的最小$T$-连接）若确定$G$的一个连通子图，则称为连通最小连接。自Middendorf和Pfeiffer证明此类嫁接图满足Seymour关于连接与$T$-割填充的极小-极大公式以来，具有连通最小连接的嫁接图备受关注；即在此类嫁接图中，最小连接的大小等于$T$-割最大填充的大小。本文提出具有连通最小连接的嫁接图的构造性表征。同时，我们设计了一个多项式时间算法，用于判定给定嫁接图是否具有连通最小连接，并在存在时输出一个实例。该算法存在两个计算瓶颈：一是计算嫁接图最小连接所需时间，二是在具有保守$\pm 1$值边权重的图中求解单源全汇最短路径问题所需时间。因此，算法的时间复杂度为$O(n(m + n\log n) )$。在非稠密图情形下，本算法改进了Seb\H{o}和Tannier基于图度量理论成果所提出的该问题时间复杂度上界。