Covering All My Bases: Analyzing Unique Efficient Dominating Sets

Presenter: Isaac Reiter, Kutztown University Abstract. Dominating sets is a very interesting topic in graph theory. A vertex is said to dominate itself and the vertices to which it is adjacent. Given a graph, a dominating set D of this graph is a subset of the vertices such that every vertex is either in D or adjacent to a vertex in D. If none of the vertices in D are adjacent to each other and if every vertex not in D is adjacent to exactly one vertex in D, then D is an efficient dominating set. In other words, an efficient dominating set ensures that every vertex is only dominated by one vertex in the dominating set. Furthermore, if the graph in question only has one efficient dominating set, then D is a unique efficient dominating set. In order to analyze this topic, the presenter and his professor developed numerous ways of creating larger graphs with unique efficient dominating sets from smaller graphs. In this talk, the presenter will give a proof of one such method.