For which graphs is the DFS tree always a path?
This is equivalent to the property that you can construct a Hamiltonian path by greedily taking an arbitrary edge at every vertex. Searching for greedy Hamiltonian path turned up: Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests, Tankusa and Tarsib, doi:10.1016/j.disc.2006.09.031, which points to Randomly Traceable Graphs, Chartrand and Kronk, SIAM J. Appl. Math., 16(4), 696–700, doi:10.1137/0116056 as characterising these graphs as precisely the graphs you mention in the question.