$x(\delta(i)) = 2$, when $OPT_{LP}(G)=n$.

For the rest of Section \[sec:lpn\], let $T$ denote a greedy DFS tree in which all 1-edges are tree edges.

\[lem:two-edge\] If $OPT_{LP}(G)=n$, $G$ is subquartic and each back edge $(i,j) \in
B(T)$ is assigned value $f(i,j) = 1/2$, then each vertex in $T_{int}$ is satisfied by $f$.

Suppose we set $g(i,j) = x_{ij}$ for each back edge. Then by the cut argument of Mömke and Svensson, each tree cut is satisfied by $g$, because $LP(G) = n$ and each vertex $v \in V$ has $x(\delta(v))=2$. Since there are no 1-edges in the set of back edges, this implies that each tree cut must in fact be covered by at least two edges. Thus, setting $f(i,j) = 1/2$ results in each tree cut being satisfied by $f$.

\[lem:three-edge\] If $OPT_{LP}(G)=n$ and $G$ is subquartic, setting $f(i,j) = 1/2$ for each edge $(i,j) \in B(T)$ yields a circulation with cost zero.

This follows from the fact that each vertex in $T_{int}$ has in-degree at most two and therefore the total $f$-value coming into a vertex is at most one. Thus, the circulation value is zero. (Note that the root can contribute $1/2$ to the circulation, but 