Geodetic graphs that arise as nontrivial underlying stress graphs

Characterize the geodetic graphs G that are the underlying stress graphs of graphs not isomorphic to G.

Background

For a graph G, the paper defines an underlying graph GS whose edges correspond to vertex pairs having stress interval consisting only of the two endpoints. The authors show that a connected graph is isomorphic to its underlying stress graph exactly when it is geodetic. They observe that some geodetic graphs, such as trees, cannot be underlying graphs of nonisomorphic graphs, whereas certain block graphs can. This motivates the explicit characterization problem stated in the conclusion.

References

There exist geodetic graphs–such as trees–that cannot serve as the underlying graph of any other graph. In contrast, some block graphs can appear as the underlying graph of other graphs. For instance, a block graph formed by amalgamating two K4 graphs at a single vertex also serves as the underlying graph of a graph obtained by amalgamating two C4 cycles at a single vertex. This observation leads us to pose the following problem. Problem 2. Characterize geodetic graphs G that are the underlying graphs of a graph not isomorphic to G.

— On the stress transit function  (2502.09153 - Anil et al., 13 Feb 2025) in Problem 2, Section 5, page 15