2-vertex connected. We can instead use a fractional function $b:B(T) \rightarrow [0,1]$ to obtain an upper bound on the cost of such a 2-vertex connected subgraph using the cost function .

One can verify that all internal vertices being satisfied by $b$ corresponds to the edges with unit demand in the network $C(G,T)$ having their demands satisfied and that, for subquartic graphs, the cost function equals the cost function used in [@DBLP:conf/focs/MomkeS11] to upper bound the cost of a circulation in $C(G,T)$.

Although finding $b$-values for the back edges that satisfy all the vertices in $T_{int}$ is equivalent to finding a feasible circulation of $C(G,T)$, and we could have stuck to the notation presented in [@DBLP:conf/focs/MomkeS11], we believe our notation results in a clearer presentation of our main theorems.

Subquartic graphs: $OPT_{LP}(G) = n$ {#sec:lpn}
====================================

We now show that in the special case when $OPT_{LP}(G)=n$ and $G$ is subquartic, there is a circulation with cost zero. Note that if $|E|
= n$, then each edge in $E$ must have $x$-value 1. Thus, $G$ is a Hamiltonian cycle. If $|E| > n$, then we can show that we can find a greedy DFS tree $T$ for $G$ such that each 