Consider the following graph, Among the following sequences: (I) a b e g h f…

To determine valid depth-first traversals, we must check if each sequence follows a valid DFS path—moving from one vertex to an adjacent unvisited vertex, and backtracking only when necessary. The graph has vertices a, b, e, f, g, h with edges connecting them.

Sequence I (a b e g h f): Starts at a, moves to b, then to e. From e, it can go to g (via edge e-g), then h (g-h), and finally f. This is a valid DFS path.

Sequence II (a b f e h g): Starts a→b→f. However, there is no edge between f and e in the graph, so moving from f to e violates DFS connectivity. This sequence is invalid.

Sequence III (a b f h g e): Starts a→b→f, then to h (via f-h), then to g (h-g), and finally to e. This path is valid as all edges exist.

Sequence IV (a f g h b e): Starts a→f→g→h. After reaching h, it backtracks to b (via h-b), then to e. This path is valid as all edges are present.

Thus, sequences I, III, and IV are valid depth-first traversals. Sequence II is invalid due to the missing edge f-e.