Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests

There are various greedy schemas to construct a maximal path in a given input graph. Associated with each such schema is the family of graphs where it always results a path of maximum length, or a Hamiltonian path/cycle, if there exists one. Considerable amount of work has been carried out, regarding the characterization and recognition problems of such graphs. We present here a systematic listing of previous results of this type and fill up most of the remaining empty entries.