The structure of minimally 2-subconnected graphs

A graph $G$ with at least $2k$ vertices is called $k$-subconnected if, for any $2k$ vertices in $G$, there are $k$ independent paths $P_1,P_2,\cdots ,P_k$ joining the $2k$ vertices in pairs. A graph $G$ is minimally 2-subconnected if $G$ is $2$-subconnected and $G-e$ is not $2$-subconnected for any edge e in G. The concept of $k$-subconnected graphs is introduced in the research of matching theory, and this concept has been found to be related with connectivity of graphs. It is of theorectical interests to characterize the structure of minimally $k$-subconnected graphs. In this paper, we characterize the structure of minimally $2$-subconnected graphs.