Is Hamiltonian path NP-hard on graphs of diameter 2? Let $G$ be a graph of diameter 2 ($\forall u,v\in V: d(u,v)\leq2$).


  Can we decide if $G$ has Hamiltonian path in poly time? What about digraphs?




Perhaps some motivation is in place:

the question arises from Dirac's theorem which states that if $\forall v\in V:d(v)\geq \frac{n}{2}$ then the graph is Hamiltonian, as well as it's generalizations (the Ghouila-Houri theorem and the result from Bang-Jensen and Gutin's book).

I've shown here that these degree requirements imply that the graph has diameter 2, and was wondering if such graphs can be decided without the degree requirements (strong gut feeling: No).
Assuming you are talking about directed graphs, the work of Füredi et al. [1] shows that such graphs of n vertices and diameter 2 have at least ~n log (n) edges, or an average degree of ~log(n).  I am aware of no subsequent result tightening this result, and the authors themselves suggest that they think this result is tight; this is quite far from the known sufficient conditions for Hamiltonian paths, which are based on a roughly linear vertex degree.

As a result, this appears to be an open and difficult problem that would require some significant research for a solution.

[1] Füredi, Zoltán, et al. "Minimal oriented graphs of diameter 2." Graphs and Combinatorics 14.4 (1998): 345-350.