text: We commence the study of domination in the incidence graphs of combinatorial designs. Let @math be a combinatorial design and denote by @math the domination number of the incidence (Levy) graph of @math . We obtain a number of results about the domination numbers of various kinds of designs. For instance, a finite projective plane of order @math , which is a symmetric @math -design, has @math . We also show that for any symmetric @math -design it holds that @math . We study at depth the domination numbers of Steiner systems and in particular of Steiner triple systems. We show that a @math has @math and also obtain a number of upper bounds. The tantalizing conjecture that all Steiner triple systems on @math vertices have the same domination number is proposed and is verified up to @math . The structure of minimal dominating sets is also investigated, both for its own sake and as a tool in deriving lower bounds on @math . Finally, a number of open questions are proposed.
  @cite_4: Switching is a local transformation that when applied to a combinatorial object gives another object with the same parameters. It is here shown that the cycle switching graph of the 11 084 874 829 isomorphism classes of Steiner triple systems of order 19 as well as the cycle switching graph of the 1 348 410 350 618 155 344 199 680 000 labeled such designs are connected. In addition to giving an understanding of the multitude of Steiner triple systems—at least for order 19 but perhaps also generally—this work also presents an algorithm for testing connectedness of large implicit graphs and brings forward a benchmark instance for such algorithms.
  @cite_8: The graph-theoretic study of combinatorial chessboard problems can be extended to the study of line graphs of graphs of combinatorial designs. In particular, the determination of optimal placements of rooks on a chessboard corresponds to the determination of domination parameters of graphs of block designs. The determination of one such parameter, the independence number, is shown to follow directly from classical results in design theory. Additionally, the domination number of graphs of finite projective planes is discussed.
summary: Laskar and Wallis in @cite_8 have obtained some results about the domination number of the line graph of @math , motivated by combinatorial chessboard problems. However, the domination number of @math itself was not considered by them. The recent thesis of H ' e ger @cite_4 is probably closeset in spirit to the present work: H ' e ger constructs dominating sets of a particular form in projective planes.