cut corresponding to one of these outgoing edges, the total $x$-value of the edges that cover this cut must be at least $1$, since this set of edges plus the tree edge forms a cut in the graph (which has $x$-value at least 2 by the constraints in $LP(G)$). Since the tree edge has $x$-value at most 1, the edges that cover the cut must have value at least 1. This argument can be applied to the tree cut corresponding to each of the three outgoing edges. So when $v$ has three outgoing tree edges, then $v$ is satisfied by $x$.

Now let us consider the case when $v$ has two outgoing tree edges. In a subquartic graph, there is at most one incoming back edge into a branch vertex $v$. This back edge comes from the subtree connected to one of the outgoing tree edges. It may be the case that the tree cut corresponding to this tree edge is not satisfied by $x$. However, in this case, consider the other outgoing tree edge that is connected to a subtree from which there is no back edge connected directly to $v$. If we remove this outgoing tree edge, the 