Realizing block-graph intersection structures

Determine whether, for every block graph G (that is, a graph whose every block is a clique), there exists a harmonic-uniform graph G' whose block-intersection graph is isomorphic to G.

Background

The paper constructs harmonic-uniform graphs with a single cut-vertex and diameter 4, as well as further examples with a unique cut-edge and larger diameter. These constructions are based on gluing blocks and demonstrate that harmonic-uniformity can coexist with nontrivial block structure.

The authors note that the known constructions have particular structural and symmetry constraints, motivating the broader question of whether arbitrary block-intersection patterns arising from block graphs can be realized by harmonic-uniform graphs. The problem asks specifically for existence while preserving the prescribed intersection graph of blocks.

References

Using AI-driven approach by chatGPT, it is possible to construct also harmonic-uniform graphs with diameter greater than 6 that contain a single cut-edge; the fact that such graphs are very large, together with the observation that all known harmonic-uniform graphs have at most two cut-vertices, raises several open questions:

Given a block-graph $G$ (that is, a graph whose every block is a clique), does there exist a harmonic-uniform graph $G'$ such that the intersection graph of the blocks of $G'$ is isomorphic to $G$?

— On harmonic centers of graphs  (2609.31295 - Madaras et al., 25 Sep 2026) in Section 4, Concluding remarks, first Problem environment