  The domination number $\gamma(G)$ of a graph $G$, its exponential domination
number $\gamma_e(G)$, and its porous exponential domination number
$\gamma_e^*(G)$ satisfy $\gamma_e^*(G)\leq \gamma_e(G)\leq \gamma(G)$. We
contribute results about the gaps in these inequalities as well as the graphs
for which some of the inequalities hold with equality. Relaxing the natural
integer linear program whose optimum value is $\gamma_e^*(G)$, we are led to
the definition of the fractional porous exponential domination number
$\gamma_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we
show $\gamma_{e,f}^*(T)=\frac{n+2}{6}$ and $\gamma_e(T)\leq
2\gamma_{e,f}^*(T)$. We characterize the two classes of subcubic trees $T$ with
$\gamma_e(T)=\gamma_{e,f}^*(T)$ and $\gamma(T)=\gamma_e(T)$, respectively.
Using linear programming arguments, we establish several lower bounds on the
fractional porous exponential domination number in more general settings.
