are the vertices that can contribute to the cost of $C(G,T)$. The root can also contribute a negligible value of either one or two to the cost of $C(G,T)$. For the sake of simplicity, we ignore the contribution of the root in most of our calculations.

[**Fact.** ]{}\[exp:bound\] The number of expensive vertices is bounded as follows: $|T_{exp}| \leq n/2$.

A [*branch*]{} vertex in $T$ is a vertex with at least two outgoing tree edges.

Note that the root of $T$ is not be a branch vertex, since $G$ is 2-vertex connected.

\[lem:bex\] If $G$ is subquartic, then a branch vertex in $T_{int}$ is not expensive.

In a graph with vertex degree at most four, a branch vertex can have at most one incoming back edge and therefore cannot be expensive.

A [*tree cut*]{} is the partition of the vertices of the tree $T$ induced when we remove an edge $(u,v) \in T$.

For each edge $(i,j) \in B(T)$, let $b(i,j) \leq 1$ be a non-negative value.

Consider a tree cut corresponding to edge $(u,v) \in T$ and remove all back edges $(w,u) \in B(T)$, where $w$ belongs to the subtree of $v$ in $T$. We say that the remaining 