The Goddard-Henning graph, illustrated above in several embeddings, is the 9-node planar graph of graph
diameter 2 having domination number . It was first constructed by MacGillivray
and Seyffarth (1996) and subsequently proved (Goddard and Henning 2002, Henning and
Yeo 2013, pp. 55-56) to be the unique diameter-2
planar graph with ; all other diameter-2
planar graphs have domination
number at most 2.

Goddard, W. Henning, M. A. "Domination in Planar Graphs with Small Diameter." J. Graph Th.40, 1-25, 2002.Henning,
M. A. and Yeo, A. Total
Domination in Graphs. New York: Springer, pp. 55-56, 2013.MacGillivray,
G. and Seyffarth, K. "Domination Numbers of Planar Graphs." J. Graph
Th.22, 213-219, 1996.