  For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a
$2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over
$3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every
$3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph
that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally,
every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a
$2$-regular subgraph that omits at most $\max\{0,\lfloor
(3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the
extremal multigraphs.
