Maximal bipartite graphs with a unique minimum dominating set
Garrison Koch, Darren Narayan
TL;DR
The paper investigates the maximal number of edges in bipartite graphs of order $n$ with domination number $\gamma$ that admit a unique minimum dominating set. It introduces a bipartite bound $m(n,\gamma)$ and provides explicit extremal constructions $\mathcal{B}_{n,\gamma}$ that achieve this bound, establishing tightness in general. It proves the bound for the important cases $\gamma=2$ and $n=3\gamma$, and discusses the role of perfect domination in these extremal graphs. The results illustrate how bipartiteness constrains edge counts relative to Fischermann's bound for general graphs and supply multiple constructions that realize the maximal size while maintaining unique domination.
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 $γ= 2$ \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 $γ= 2$ and $n = 3γ$ 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.