  We propose the conjecture that the domination number $\gamma(G)$ of a
$\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge
domination number $\gamma_e(G)$, which coincides with the domination number of
its line graph. We prove that $\gamma(G)\leq \left(1+\frac{2(\Delta-1)}{\Delta
2^{\Delta}}\right)\gamma_e(G)$ for general $\Delta\geq 1$, and $\gamma(G)\leq
\left(\frac{7}{6}-\frac{1}{204}\right)\gamma_e(G)$ for $\Delta=3$. Furthermore,
we verify our conjecture for cubic claw-free graphs.
