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.

Solution 1:

I really did not expect this to have a term, but according to Wikipedia:

• 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 $G$.

There is, however, another notion of cone graph which refers to a more specific graph (which is only sometimes a cone in the above terminology).