Minimum $T$-Joins and Signed-Circuit Covering

Let G be a graph and 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)| − 1 2 |E( Ĝ )| edges where Ĝ is the maximum bidegeless subgraph of G. Further, we are able to use this result to show that every flow-admissible signed graph (G, σ) has a signed-circuit cover with length at most 19 6 |E(G)|. Particularly, a 2-edge-connected signed graph (G, σ) with even negativeness has a signed-circuit cover with length at most 8 3 |E(G)|.