Classification of 2-connected diameter-two graphs with fewer lines than vertices

Classify all 2-connected graphs of diameter two that have fewer lines than vertices, determining whether the known examples—$H_6$, $H_8$, the house $H_5$, and the graph in Theorem 2.1 of the cited work—are the only such graphs.

Background

The paper proves that every connected locally connected graph of diameter two with fewer than n lines is one of $K_{1,2,2}$, $K_{2,2,2}$, or $K_{2,2,2,2}$. It then observes that this classification does not extend immediately to all 2-connected graphs: several additional examples are known, including the graphs $H_6$, $H_8$, and the house $H_5$. The authors explicitly state that these are the only known examples and express a belief that no others exist, leaving the general classification unresolved.