Incidence-free sets and edge domination in incidence graphs

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