It is well known that every sufficiently large connected graph has, as an induced subgraph, $K_n$, $K_{1,n}$, or an $n$-vertex path. A 2023 paper of Allred, Ding, and Oporowski identified a set of unavoidable induced subgraphs of sufficiently large $2$-connected graphs. In this paper, we establish a dual version of this theorem by focusing on the minors obtained by contracting cycles.