I'm defining "Region Free Subgraphs" as follows:
Let $G_e$ be a planar embedding of a graph $G$. The Region Free Subgraph of $G_e$ is the subgraph of $G$ made up of all edges in $G$ that are not a member of a bounded face in $G_e$.
$G_e$ (region free subgraph in orange)
Is there another term for this type of subgraph already in use, or should I keep calling them "Region Free Subgraphs"?
Update
Unless I get a better answer, will use the term “Bridge Subgraphs” based on the feedback I received from @MathewDaly