Edges in and out of cycles and their relation to connectivity of the graph
Yes.  Let $G$ be a graph, and suppose that the edge $ij\in E(G)$ is not on any cycle.  This implies that every path $P$ from $i$ to $j$ in $G$ uses the edge $ij$, since, otherwise, $P\cup \{ij\}$ is a cycle in $G$.  In particular, removing $ij$ from $G$ disconnects $i$ from $j$.

These kinds of edges are called bridges.