A maximally nonhamiltonian graph is a nonhamiltonian graph 
for which 
is Hamiltonian for each edge 
in the graph complement
of 
,
i.e., every two nonadjacent vertices are endpoints of a Hamiltonian
path.
Since an edge added between two disconnected components of a disconnected graphs is a bridge, and after crossing a bridge in one direction, a Hamiltonian cycle cannot "cross back" in the other, it can be seen that only connected nonhamiltonian graphs can be maximally nonhamiltonian.
The status of the path graph 
is debatable since it is nonhamiltonian but has empty graph
complement, but by a strict interpretation that "all 0 of its complement's edges
result in a Hamiltonian graph," it should be considered maximally nonhamiltonian.
The numbers of maximally nonhamiltonian graphs on 
, 2, ... vertices are 0, 1, 1, 1, 3, 3, 7, 9, 18, 31, ...
(OEIS A185306), the first few of which are
illustrated above. Larger examples the Coxeter graph,
flower snarks 
for odd 
, Petersen graph, and
Tietze's graph (Clark and Entringer 1983).
