# Depth First Search¶

Depth First Search is one of the main graph algorithms.

Depth First Search finds the lexicographical first path in the graph from a source vertex $u$ to each vertex. Depth First Search will also find the shortest paths in a tree (because there only exists one simple path), but on general graphs this is not the case.

The algorithm works in $O(m + n)$ time where $n$ is the number of vertices and $m$ is the number of edges.

## Description of the algorithm¶

The idea behind DFS is to go as deep into the graph as possible, and backtrack once you are at a vertex without any unvisited adjacent vertices.

It is very easy to describe / implement the algorithm recursively: We start the search at one vertex. After visiting a vertex, we further perform a DFS for each adjacent vertex that we haven't visited before. This way we visit all vertices that are reachable from the starting vertex.

For more details check out the implementation.

• Find any path in the graph from source vertex $u$ to all vertices.

• Find lexicographical first path in the graph from source $u$ to all vertices.

• Check if a vertex in a tree is an ancestor of some other vertex:

At the beginning and end of each search call we remember the entry and exit "time" of each vertex. Now you can find the answer for any pair of vertices $(i, j)$ in $O(1)$: vertex $i$ is an ancestor of vertex $j$ if and only if $\text{entry}[i] < \text{entry}[j]$ and $\text{exit}[i] > \text{exit}[j]$.

• Find the lowest common ancestor (LCA) of two vertices.

• Topological sorting:

Run a series of depth first searches so as to visit each vertex exactly once in $O(n + m)$ time. The required topological ordering will be the vertices sorted in descending order of exit time.

• Check whether a given graph is acyclic and find cycles in a graph. (As mentioned above by counting back edges in every connected components).

• Find strongly connected components in a directed graph:

First do a topological sorting of the graph. Then transpose the graph and run another series of depth first searches in the order defined by the topological sort. For each DFS call the component created by it is a strongly connected component.

• Find bridges in an undirected graph:

First convert the given graph into a directed graph by running a series of depth first searches and making each edge directed as we go through it, in the direction we went. Second, find the strongly connected components in this directed graph. Bridges are the edges whose ends belong to different strongly connected components.

## Classification of edges of a graph¶

We can classify the edges of a graph, $G$, using the entry and exit time of the end nodes $u$ and $v$ of the edges $(u,v)$. These classifications are often used for problems like finding bridges and finding articulation points.

We perform a DFS and classify the encountered edges using the following rules:

If $v$ is not visited:

• Tree Edge - If $v$ is visited after $u$ then edge $(u,v)$ is called a tree edge. In other words, if $v$ is visited for the first time and $u$ is currently being visited then $(u,v)$ is called tree edge. These edges form a DFS tree and hence the name tree edges.

If $v$ is visited before $u$:

• Back edges - If $v$ is an ancestor of $u$, then the edge $(u,v)$ is a back edge. $v$ is an ancestor exactly if we already entered $v$, but not exited it yet. Back edges complete a cycle as there is a path from ancestor $v$ to descendant $u$ (in the recursion of DFS) and an edge from descendant $u$ to ancestor $v$ (back edge), thus a cycle is formed. Cycles can be detected using back edges.

• Forward Edges - If $v$ is a descendant of $u$, then edge $(u, v)$ is a forward edge. In other words, if we already visited and exited $v$ and $\text{entry}[u] < \text{entry}[v]$ then the edge $(u,v)$ forms a forward edge.

• Cross Edges: if $v$ is neither an ancestor or descendant of $u$, then edge $(u, v)$ is a cross edge. In other words, if we already visited and exited $v$ and $\text{entry}[u] > \text{entry}[v]$ then $(u,v)$ is a cross edge.

Theorem. Let $G$ be an undirected graph. Then, performing a DFS upon $G$ will classify every encountered edge as either a tree edge or back edge, i.e., forward and cross edges only exist in directed graphs.

Suppose $(u,v)$ is an arbitrary edge of $G$ and without loss of generality, $u$ is visited before $v$, i.e., $\text{entry}[u] < \text{entry}[v]$. Because the DFS only processes edges once, there are only two ways in which we can process the edge $(u,v)$ and thus classify it:

• The first time we explore the edge $(u,v)$ is in the direction from $u$ to $v$. Because $\text{entry}[u] < \text{entry}[v]$, the recursive nature of the DFS means that node $v$ will be fully explored and thus exited before we can "move back up the call stack" to exit node $u$. Thus, node $v$ must be unvisited when the DFS first explores the edge $(u,v)$ from $u$ to $v$ because otherwise the search would have explored $(u,v)$ from $v$ to $u$ before exiting node $v$, as nodes $u$ and $v$ are neighbors. Therefore, edge $(u,v)$ is a tree edge.

• The first time we explore the edge $(u,v)$ is in the direction from $v$ to $u$. Because we discovered node $u$ before discovering node $v$, and we only process edges once, the only way that we could explore the edge $(u,v)$ in the direction from $v$ to $u$ is if there's another path from $u$ to $v$ that does not involve the edge $(u,v)$, thus making $u$ an ancestor of $v$. The edge $(u,v)$ thus completes a cycle as it is going from the descendant, $v$, to the ancestor, $u$, which we have not exited yet. Therefore, edge $(u,v)$ is a back edge.

Since there are only two ways to process the edge $(u,v)$, with the two cases and their resulting classifications outlined above, performing a DFS upon $G$ will therefore classify every encountered edge as either a tree edge or back edge, i.e., forward and cross edges only exist in directed graphs. This completes the proof.

## Implementation¶

vector<vector<int>> adj; // graph represented as an adjacency list
int n; // number of vertices

vector<bool> visited;

void dfs(int v) {
visited[v] = true;
for (int u : adj[v]) {
if (!visited[u])
dfs(u);
}
}

This is the most simple implementation of Depth First Search. As described in the applications it might be useful to also compute the entry and exit times and vertex color. We will color all vertices with the color 0, if we haven't visited them, with the color 1 if we visited them, and with the color 2, if we already exited the vertex.

Here is a generic implementation that additionally computes those:

vector<vector<int>> adj; // graph represented as an adjacency list
int n; // number of vertices

vector<int> color;

vector<int> time_in, time_out;
int dfs_timer = 0;

void dfs(int v) {
time_in[v] = dfs_timer++;
color[v] = 1;