Cuts on a Circle Beta

Let $G$ be a connected simple graph with $2n$ vertices, where $n\ge 2$.

For two distinct vertices $u$ and $v$, let $c(u,v)$ be the minimum number of edges that must be removed to disconnect $u$ from $v$.

For each vertex $v$, consider the $2n-1$ values $c(v,u)$ over all other vertices $u$. Let $b(v)$ be the sum of the $n$ smallest values.

Arrange all vertices around a circle. The cost of an arrangement is the sum of $c(u,v)$ over all consecutive pairs, including the last and first vertices. Consecutive vertices do not need to be adjacent in $G$.

Prove that the minimum possible cost is

$$2\max_{v\in V(G)} b(v).$$