The $k$-conversion number of regular graphs

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 $m$.