# Graphs with domination number $\gamma(G)=1$.

Let $$\gamma(G)$$ be the domination number of a graph $$G$$. I was wondering if there is special terminology for graphs with $$\gamma(G)=1$$? Examples of graphs with $$\gamma(G)=1$$ include $$K_{1,n}$$ and Wheel graphs.

• In other words, graphs obtained by adding edges to a star graph? Commented Sep 3, 2021 at 12:42
• @HagenvonEitzen, exactly! Subgraphs of a complete graph on $n$ vertices that have a star graph as a subgraph. Commented Sep 3, 2021 at 12:57

• If $$G$$ has a dominating set $$\{v\}$$, $$v$$ is called a universal vertex or dominating vertex;
• $$G$$ itself is called a cone when it has a dominating vertex.
Wikipedia is iffy about calling $$v$$ the apex of the cone, because this conflicts with other terminology, but apparently this is also a thing sometimes.
We can be more precise than calling $$G$$ a cone. In sources like this paper, for any graph $$G$$, we can construct the cone of $$G$$, denoted $$CG$$, by adding a new vertex adjacent to all vertices of $$G$$.