# How to construct a planar graph (or a class of planar graphs) with minimum degree 5 of diameter 2?

A subset $$D$$ of the vertices of a graph G is called a dominating set if every vertex of $$G - D$$ is adjacent to a vertex of $$D$$, and the domination number of $$G$$, denoted $$\gamma(G)$$, is the minimum cardinality of a dominating set.

The following article proves that any planar graph with diameter 2 has domination number at most 3.

• MacGillivray G, Seyffarth K. Domination numbers of planar graphs[J]. Journal of Graph Theory, 1996, 22(3): 213-229.

As part of the proof, the authors also showed that planar graphs with minimum degree 5 of diameter two have domination number at most three. But the author did not construct the corresponding example graph.

My question is

• whether a planar graph with minimum degree of 5 having diameter 2 exists. (Of course, it is better to construct a class of there graphs)

Another question is

• whether a planar graph with minimum degree of 5 of diameter 2 having domination number three exists. (Of course, it is better to construct a class of there graphs)

I am aware that there are some planar graphs with minimum degree 5 of diameter 3 (see an icosahedron graph as a example).

Or we can see that some planar graphs with minimum degree 4 of diameter 2 (see an octahedral graph as a example).

There is no planar graph of diameter two with minimal degree $$5$$. This follows from Seyffarth theorem, which is formulated as follows.
Let $$G$$ be a maximal planar graph with maximum degree $$\Delta$$ and diameter two. Then the minimum degree $$\delta<5$$ (Theorem 2), p.623.