Walks, Paths and Connectivity · Superchallenge problem

A graph has a central path $0\text{-}1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}6$, with a triangle attached at vertex $0$ (extra vertices $7,8$) and a triangle attached at vertex $6$ (extra vertices $9,10$). Among the $6$ bridges of this graph, find the smallest possible difference between the sizes of the two pieces that removing a single bridge creates.