SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1192-1204, January 2020.
Let $G$ be a graph with edge set $E(G)$ and vertex set $V(G)$, and let $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bridgeless subgraph of $G$, and the bound is optimal. Further, we are able to use this result to show that every flow-admissible signed graph $(G,\sigma)$ has a signed-circuit cover with length at most $\frac{19} 6 |E(G)|$. Particularly, a 2-edge-connected signed graph $(G,\sigma)$ with even negativeness has a signed-circuit cover with length at most $\frac 8 3 |E(G)|$.

中文翻译：

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最低$ T $联接和签名电路覆盖

SIAM离散数学杂志，第34卷，第2期，第1192-1204页，2020年1月。
设$ G $为边集为$ E（G）$且顶点为$ V（G）$的图，令$ T $为基数为$ G $的顶点子集。$ G $的$ T $联接是边的子集$ J $，这样，当且仅当顶点属于$ T $时，$ G $的顶点才以$ J $中的奇数个边入射。最小$ T $ -joins在组合优化中有许多应用。在本文中，我们显示了连通图$ G $的最小$ T $ -join最多具有$ | E（G）|-\ frac 1 2 | E（\ widehat {\，G \，}）| $ edges，其中$ \ widehat {\，G \，} $是$ G $的最大无桥子图，并且边界是最佳的。此外，我们可以使用此结果显示每个允许流量的有符号图$（G，\ sigma）$都有一个带符号的电路覆盖层，其长度最大为$ \ frac {19} 6 | E（G）| $ 。特别是2边连接的有符号图$（G，