What is the time complexity of a depth-first search (DFS) on a graph with V vert

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Question: What is the time complexity of a depth-first search (DFS) on a graph with V vertices and E edges?

Options:

Correct Answer: O(V + E)

Solution:

DFS visits each vertex and edge once, leading to a time complexity of O(V + E).

What is the time complexity of a depth-first search (DFS) on a graph with V vertices and E edges?

What is the time complexity of a depth-first search (DFS) on a graph with V vertices and E edges?

Step 1: Understand that a graph consists of vertices (V) and edges (E). Vertices are the points, and edges are the connections between them.
Step 2: Recognize that during a depth-first search (DFS), we explore each vertex and each edge in the graph.
Step 3: Note that when we visit a vertex, we may also traverse its edges to reach other vertices.
Step 4: Since we visit each vertex once, this contributes O(V) to the time complexity.
Step 5: Similarly, since we traverse each edge once, this contributes O(E) to the time complexity.
Step 6: Combine the contributions from vertices and edges to get the total time complexity: O(V) + O(E).
Step 7: Conclude that the overall time complexity of DFS is O(V + E).

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