How is a depth-first search implemented in a graph?

A depth-first search in a graph is implemented by traversing through the graph deeply before exploring neighbours.

Depth-first search (DFS) is a fundamental algorithm in graph theory, used to traverse or search through a graph in a systematic manner. The process begins at a specific node (often the root of the tree, if such a node exists), then explores as far as possible along each branch before backtracking.

To implement DFS, you would typically use a stack data structure. The algorithm starts at a chosen node (often the root), pushes it onto the stack, and marks it as visited. It then examines the top node of the stack and checks for any unvisited nodes connected to it. If it finds an unvisited node, it pushes that node onto the stack and marks it as visited. If there are no unvisited nodes, it pops the top node off the stack. The process continues until the stack is empty.

The pseudocode for DFS is as follows:

```
DFS(graph, root):
create a stack S and push root into S
create a set V (visited nodes)
while S is not empty:
pop the top node N from S
if N is not in V:
add N to V
push all neighbours of N into S
```

DFS is a recursive algorithm, meaning it calls itself to solve sub-problems. This is evident in the way it dives deep into a branch before backtracking. This makes DFS well-suited for problems where you want to go as deep as possible into a graph, such as solving mazes or finding connected components in a graph.

However, DFS is not suitable for all problems. For instance, in a weighted graph, DFS does not necessarily find the shortest path to a node. For such problems, other algorithms like Dijkstra's or A* would be more appropriate.

In conclusion, DFS is a powerful tool for traversing and searching graphs, but like all algorithms, it has its strengths and weaknesses and should be used judiciously.