Unavoidable immersions of 4- and $f(t)$-edge-connected graphs

In this paper we prove that every sufficiently large 4-edge-connected graph contains the double cycle, $C_{2,r}$, as an immersion. In proving this, we develop a new tool we call a ring-decomposition. We also prove that linear edge-connectivity implies the presence of a $C_{t,r}$ immersion in a sufficiently large graph, where $C_{t,r}$ denotes the graph obtained from a cycle on $r$ vertices by adding $(t-1)$ edges in parallel to each existing edge; this result is an edge-analogue of a result of B\"{o}hme, Kawarabayashi, Maharry, and Mojar. We then use the latter result to provide an unavoidable minor theorem for highly connected line graphs.