I really don't understand this definition from this paper which is:
$A-bridge$: if $A \subseteq V(G)$, then an $A-bridge$ of $G$ is either an edge joining two vertices of $A$ or an edge-maximal sub-graph $H$ of $G$ that does not contain an edge between two vertices of $H$ and such that there is a path between any two vertices of $H$ with all its inner vertices distinct from the vertices of A.
Another thing to add is,that all the graphs in this paper are series parallel graphs.
It would be great to clarify this definition with some pictures or reference to some good sources!
The definition given isn't clear at all, but the concept is a fairly well-known one. My guess is that an $A$-bridge is an edge of $G$ with both ends in $A$, or a subgraph $B$ of $G$ consisting of a connected component $H$ of $G-A$, together with all edges of $G$ between vertices of $H$ and vertices of $A$ (and the vertices of $A$ incident with those edges—the attachments of $B$). Hopefully you will be able to tell from context if this is correct.