21.1.7. Applications of DFS Numbers

So, many useful features about the graph can actually be recorded by can be extracted by recording the order in which DFS which is vertices. So, for this we argument DFS, we something called numbering.

In Depth First Search (DFS), we can gather important information from the way we traverse the graph. This is achieved through something known as DFS numbering. Essentially, as we visit each vertex during the DFS process, we also record two specific values: the 'pre-number' and the 'post-number'. The pre-number is assigned when we first enter the vertex, while the post-number is recorded just before we finish exploring that vertex and its neighbors. This numbering helps us track the order of exploration and relationships between vertices.

Imagine you're navigating a maze. As you enter a room, you mark the time you entered (pre-number). Once you've checked all possible exits from that room and are ready to leave, you note down the time you exited (post-number). By keeping track of these times, you can figure out which paths you took, how long you spent in each room, and even identify any dead ends you encountered.

So, we start by initializing this counter to 0, now whenever I invoke DFS of i, the first thing I do is assign the current counter value to something call the 3 number of i. So, this is the number of the counter before the DFS of i actually started, and then i increment the count.

To implement DFS numbering, we first start with a counter initialized to zero. Every time we call DFS for a vertex 'i', we take the current value of this counter and assign it to the vertex's pre-number. This pre-number allows us to see when we entered that vertex. Then, we increment the counter for the next vertex we encounter. When we finish exploring all possible connections from vertex 'i', we then assign a post-number to this vertex before we leave it, which indicates the time of completion for that vertex’s exploration.

Think of a time clock at a workplace where every employee must check in and out. When an employee enters (pre-number), they punch in their time. When they leave (post-number), they punch out. This way, the company can keep track of when employees were present, how long they stayed, and even find out who left first or last based on these time records.

So, we can find out thinks like whether graphs are the cycle. So, this is a cycle. So, whether a graph has a loop like this or we can find out whether there are go to see is like this.

DFS numbers are quite powerful because they help us analyze the graph's structure beyond simple traversal. For instance, by examining the pre and post numbers, we can identify whether a graph contains cycles. A cycle occurs when a vertex can be revisited without backtracking to an earlier vertex. Additionally, we can learn about critical vertices (or cut vertices) that, if removed, would disconnect the graph. This kind of information is crucial for understanding the robustness and connectivity of networks.

Consider a city's public transportation map as a graph. By analyzing the routes (edges) and the bus stops (vertices), we can determine if any route loops back on itself (cycle) or if a specific bus stop is essential for connecting two neighborhoods. If that bus stop were to close, the neighborhoods would be disconnected, making the analysis of such points vital for efficient public transport planning.

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