I don't have access to the cited paper but if the term bridge 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 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.