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