Dominating sets in Kneser graphs

Abstract

This thesis investigates dominating sets in Kneser graphs as well as a selection of other topics related to graph domination. Dominating sets in Kneser graphs, especially those of minimum size, often correspond to interesting combinatorial incidence structures.

We begin with background on the dominating set problem and a review of known bounds, focusing on algebraic bounds. We then consider this problem in the Kneser graphs, discussing basic results and previous work. We compute the domination number for a few of the Kneser graphs with the aid of some original results. We also investigate the connections between Kneser graph domination and the theory of combinatorial designs, and introduce a new type of design that generalizes the properties of dominating sets in Kneser graphs. We then consider dominating sets in the vector space analogue of Kneser graphs. We end by highlighting connections between the dominating set problem and other areas of combinatorics. Conjectures and open problems abound.