  As a natural variant of domination in graphs, Dankelmann et al. [Domination
with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce
exponential domination, where vertices are considered to have some dominating
power that decreases exponentially with the distance, and the dominated
vertices have to accumulate a sufficient amount of this power emanating from
the dominating vertices. More precisely, if $S$ is a set of vertices of a graph
$G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in
S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex
$u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance
between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{
v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum
order of an exponential dominating set of $G$.
  In the present paper we study exponential domination in subcubic graphs. Our
results are as follows: If $G$ is a connected subcubic graph of order $n(G)$,
then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq
\frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that
$\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least
$g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic
graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a
subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic
tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum
exponential dominating set problem is APX-hard for subcubic graphs.
