What is the complexity of traversing a graph using depth-first search (DFS)?

The complexity of traversing a graph using depth-first search (DFS) is O(V + E), where V represents the number of vertices (or nodes) in the graph and E represents the number of edges. This complexity arises because DFS visits each vertex and explores all edges connected to those vertices.

During the traversal, the algorithm starts at a selected vertex and explores as far as possible along each branch before backtracking. In the process, every vertex will be visited once, accounting for the O(V) part of the runtime. Once a vertex is reached, DFS will examine each of its adjacent edges to move to the next vertex, contributing to the O(E) part of the complexity.

Thus, the total complexity combines both aspects, leading to O(V + E). This makes the answer correct, as it reflects the operations performed regardless of whether the graph is sparse or dense. The other options do not capture the full scope of the traversal process—focusing solely on vertices or edges, or misrepresenting the relationship between them.