  A set of edges $\Gamma$ of a graph $G$ is an edge dominating set if every
edge of $G$ intersects at least one edge of $\Gamma$, and the edge domination
number $\gamma_e(G)$ is the smallest size of an edge dominating set. Expanding
on work of Laskar and Wallis, we study $\gamma_e(G)$ for graphs $G$ which are
the incidence graph of some incidence structure $D$, with an emphasis on the
case when $D$ is a symmetric design. In particular, we show in this latter case
that determining $\gamma_e(G)$ is equivalent to determining the largest size of
certain incidence-free sets of $D$. Throughout, we employ a variety of
combinatorial, probabilistic and geometric techniques, supplemented with tools
from spectral graph theory.
