A point set X in the plane is called a k-distance set if there are exactly k different distances between two distinct points in X. Let D = D(X) be the diameter of a finite set X, and let X-D = {x is an element of X : d(x, y) = D for some y is an element of X}, the diameter graph DG(X-D) of X-D is the graph with X-D as its vertices and where two vertices x, y is an element of X-D are adjacent if d(x, y) = D. We prove the set X having at most five distances with DG(X-D) = C-7 has the unique X-D = R-7, and the set X having at most six distances with DG(X-D) = C-9 has the unique X-D = R-9, and give a conjecture for k-distance set with DG(X-D) = C2k-3.
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