Minimum T-Joins and Signed-Circuit Covering
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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( G )| edges where G 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)|.
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