Incidence-free sets and edge domination in incidence graphs

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.