I don't have access to the cited paper but if the term [bridge][1] is used as in the answer of @Greg Kuperberg then other synonyms from graph theory are **isthmus**, **cut-edge**, and **cut-arc**. An important feature of such a bridge $e$ of a (possibly disconnected) graph $G$ is that if $F$ is any spanning forest of $G$, then $e$ must be an edge in $F$. Equivalently, $e$ is not contained in any cycle of $G$. If $G$ is planar then the dual edge $e^*$ in the dual graph $G^*$ is a loop (a cycle of length one).

More generally, one can study the [Tutte polynomial][2] of a matroid. In this context an element $e$ that is in every basis of a given matroid is called an **isthmus** or a **coloop**. 


  [1]: https://en.wikipedia.org/wiki/Bridge_(graph_theory)
  [2]: https://en.wikipedia.org/wiki/Matroid#Tutte_polynomial