Procedure DFS(graph, parents, sequence, i)
sequence = sequence ++ [i]
foreach j in columns(graph){
if(graph[i][j] == 1 and not(isKey(parents, j))){
parents[j] = i
parents, sequence = DFS(graph, parents, sequence, j)
}
}
return(parents, sequence)
End DFS

This pseudocode outlines a **Depth-First Search (DFS)** algorithm for traversing a graph. Here's a detailed breakdown of each component:

### Inputs:
- **graph**: A 2D array (adjacency matrix) representing the graph. If `graph[i][j]` is 1, there is an edge from node `i` to node `j`.
- **parents**: A dictionary that records the parent of each node in the DFS tree.
- **sequence**: A list that records the order in which nodes are visited during the DFS.
- **i**: The current node being explored.

### Outputs:
- **parents**: The updated dictionary showing the parent of each node in the DFS tree.
- **sequence**: The updated list showing the order in which nodes were visited.

### Procedure:
1. **Append the current node** `i` to the `sequence`.
2. **Iterate through each column** `j` of the `graph` (i.e., check all possible nodes that could be connected to `i`):
   - If there is an edge from `i` to `j` (`graph[i][j] == 1`) and `j` has not been visited yet (`not(isKey(parents, j))`):
     - Set the parent of `j` as `i`.
     - Recursively call DFS with `j` as the new current node.
3. **Return the updated** `parents` and `sequence`.

### Explanation:
- The **DFS** algorithm starts from a given node `i` and explores as far as possible along each branch before backtracking.
- **`parents`** keeps track of the DFS tree, which shows the structure of how the nodes are connected as the algorithm progresses.
- **`sequence`** provides the order of nodes visited, which is useful for understanding the traversal path.

### Example:
Consider a graph represented by an adjacency matrix:

\[
\text{graph} = 
\begin{bmatrix}
0 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 \\
\end{bmatrix}
\]

### Questions to Explore:
1. What is the difference between DFS and BFS (Breadth-First Search)?
2. How does the DFS algorithm behave in a graph with cycles?
3. What modifications would be needed for DFS to find all connected components in a graph?
4. How would you implement DFS for a graph represented by an adjacency list instead of an adjacency matrix?
5. What are the time and space complexities of the DFS algorithm?

### Tip:
Always consider edge cases, such as disconnected graphs or graphs with cycles, when implementing DFS.

Explore the Depth-First Search (DFS) algorithm applied to graph traversal using an adjacency matrix. This detailed pseudocode explanation covers how DFS navigates through nodes, updates parent relationships, and records visitation order. Ideal for computer science students and enthusiasts interested in algorithmic graph theory.

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