Minimum cardinality realizing a k-central graph

Determine, for each positive integer k, the smallest cardinality of a finite set X for which there exists a metric d on X such that the central graph of (X,d) is k-central.

Background

The problem asks for the minimum number of points needed to realize a prescribed number k=|C(X)| of central distances in a finite metric space. It is explicitly connected to the first conjecture, which concerns the special case in which the central graph is complete and k equals |X|.

The paper notes that the analogous problem for ultrametric spaces has already been solved: every k-central finite ultrametric space satisfies |X|≥2{k-1}, and equality cases have been structurally characterized in cited work. The unrestricted metric-space version remains posed as an open question.

References

Conjecture \ref{skin} is connected with the next open question. What is the smallest cardinality of the set $X$ for which there exists a metric $d\colon X\times X\to\mathbb{R}{+}$ such that the central graph of $(X,d)$ is $k$-central?

— On the Graph Corresponding to Center of Distances of Metric Space  (2610.11596 - Dovgoshey et al., 8 Oct 2026) in Problem 1, Section 4 (Expected results)