A diameter graph of a finite set 
of at least two points in Euclidean
space is the simple graph with vertex
set 
in which two distinct vertices are adjacent iff their Euclidean distance
is the generalized diameter of 
(Graham 1975, p. 166). The graph
diameter is a different quantity defined using graph
distance.
For the vertex set of a regular polygon with an odd number 
of sides, the diameter graph is the cycle
graph 
. For an even number 
of sides, it is the graph
disjoint union of 
copies of the complete graph 
, joining opposite vertices.
Indeed, the longest chords advance 
steps around the odd polygon
and 
steps around the even polygon.
Diameter graphs record the distance constraints attained with equality in the biggest little polygon problem, where the polygon diameter is fixed and the area is maximized.
Brass et al. (2005, p. 231) asked whether deleting a bounded number of vertices from a diameter graph in 
suffices to leave a 3-colorable induced
subgraph. Lopez-Campos (2026) proved that deleting at most one graph
vertex suffices. The edges in this subgraph are
inherited from the original diameter graph.
Lopez-Campos (2026) also showed that for every positive integer 
there is a finite set 
for which every subset
of smaller generalized diameter has fewer
than 
points. This construction uses a diameter graph realization
theorem of Damásdi (2026) that Lopez-Campos identifies as under review.
