A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs with a vital linkage of order 2: they are certain minors of a family of highly structured graphs.
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Wiley
Journal of Graph Theory
Networks and Optimization

Mayhew, D., Whittle, G., & van Zwam, S. (2011). The structure of graphs with a vital linkage of order 2. Journal of Graph Theory.