  Given a graph $G=(V,E)$ and a set $S_0\subseteq V$, an irreversible
$k$-threshold conversion process on $G$ is an iterative process wherein, for
each $t=1,2,\dots$, $S_t$ is obtained from $S_{t-1}$ by adjoining all vertices
that have at least $k$ neighbours in $S_{t-1}$. We call the set $S_0$ the seed
set of the process, and refer to $S_0$ as an irreversible $k$-threshold
conversion set, or a $k$-conversion set, of $G$ if $S_t=V(G)$ for some $t\geq
0$. The $k$-conversion number $c_{k}(G)$ is the size of a minimum
$k$-conversion set of $G$.
  A set $X\subseteq V$ is a decycling set, or feedback vertex set, if and only
if $G[V-X]$ is acyclic. It is known that $k$-conversion sets in $(k+1)$-regular
graphs coincide with decycling sets.
  We characterize $k$-regular graphs having a $k$-conversion set of size $k$,
discuss properties of $(k+1)$-regular graphs having a $k$-conversion set of
size $k$, and obtain a lower bound for $c_k(G)$ for $(k+r)$-regular graphs. We
present classes of cubic graphs that attain the bound for $c_2(G)$, and others
that exceed it---for example, we construct classes of $3$-connected cubic
graphs $H_m$ of arbitrary girth that exceed the lower bound for $c_2(H_m)$ by
at least $m$.
