find a subset of back edges, $S
\subseteq B(T)$, so that $T\cup S$ is 2-vertex connected and so that the following cost function has a low value: $$\begin{aligned}
\sum_{j \in T_{exp}} \max \left\{ 0, ~\left(\sum_{i:(i,j) \in S} 1
\right) -1\right\}.\end{aligned}$$ If we assign values to the edges in $B(T)$, then the only vertices that can add to the cost function are the expensive vertices, since the maximum value allowed on an edge is one. When computing the cost of the 2-vertex connected subgraph that corresponds to an integral circulation of $C(G,T)$, Mömke and Svensson use a special rule to account for the cost on a branch vertex $u$: for each outgoing edge $(u,v) \in T$, the number of incoming back edges in the form of $(w,u)$ emanating from the subtree rooted at $v$ minus one is added to the cost. However, in the case of subquartic graphs, a branch vertex is not expensive (Lemma \[lem:bex\]). Thus, in this special case, the simplified cost function in is valid.

If $b: B(T) \rightarrow \{0,1\}$ is an integral function and every internal vertex in $T$ is satisfied by $b$, then the edges with $b$-value 1 form a set $S$ such that $T+S$ is 