Maximum graphs non-Hamiltonian-connected from a vertex

A path (cycle) in a graph G is called a hamiltonian path (cycle) of G if it contains every vertex of G. A graph is hamiltonian if it contains a hamiltonian cycle. A graph G is hamiltonian-connectedif it contains a u-vhamiltonian path for each pair u, v of distinct vertices of G. A graph G is hamiltonian-connected from a vertex v of G if G contains a v-whamiltonian path for each vertex w≠v. Considering only graphs of order at least 3, the class of graphs hamiltonian-connected from a vertex properly contains the class of hamiltonian-connected graphs and is properly contained in the class of hamiltonian graphs.