back edges that cross this tree cut [*cover*]{} the cut. If the total $b$-value of the edges that cover the cut is at least 1, then we say that this tree cut is [ *satisfied by $b$*]{}.

We extend this definition to the vertices of $T$.

A vertex $v$ in $T$ is [*satisfied by $b$*]{} if for each adjacent outgoing edge in $T$, the corresponding tree cut is satisfied by $b$. On the other hand, if there is at least one adjacent outgoing edge whose corresponding tree cut is not satisfied by $b$, then the vertex $v$ is [*unsatisfied by $b$*]{}.

Mömke and Svensson define a circulation network, $C(G,T)$ (see Section 4 of [@DBLP:conf/focs/MomkeS11]), and use the cost of a feasible circulation to upper bound the length of a TSP tour in $G$. (See Lemma \[lem:MS\].)

\[lem:four\] Let $b: B(T) \rightarrow [0,1]$ and let $G$ be a subquartic graph. If each internal vertex in $T$ is satisfied by $b$, then there is a feasible circulation of $C(G,T)$ whose cost is upper bounded by the following function: $$\begin{aligned}
\sum_{j \in T_{exp}} \max \left\{ 0, ~\left(\sum_{i:(i,j) \in B(T)}
b(i,j)\right) -1\right\}.\label{cost-func-main}\end{aligned}$$

The ultimate purpose of the circulation network $C(G,T)$ in [@DBLP:conf/focs/MomkeS11] is to 