Maximal bipartite graphs with a unique minimum dominating set

Abstract

In 2003, Fischermann et al. considered the maximum size of \textit{uniquely-dominatable} graphs, graphs whose dominating number is realized only by a unique dominating set. They conjectured a size bound and provide a general graph construction that shows the bound is tight \cite{Original_Paper}. In 2010, Shank and Fraboni prove Fischermann's bound is true when $\gamma = 2$ \cite{Shank_paper}. In this paper, we observe how Fischermann's bound changes if we impart a different restriction on a graph -- bipartiteness. We conjecture a bound on the maximum number of edges possible for uniquely-dominatable bipartite graphs. We provide constructions to demonstrate this bound is tight. We prove our bipartite bound is true for the $\gamma = 2$ and $n = 3\gamma$ cases. We also discuss perfect domination and how it relates to our extremal graph constructions. We provide constructions that meet both Fischermann's bound for all graphs and our bound for bipartite graphs respectively, both of which are perfectly dominated.