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<title>Strongly Connected Components</title>
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<h1>Strongly Connected Components</h1>
<p class="editor-contents__short-description">In this tutorial, you will learn how strongly connected components are formed. Also, you will find working examples of Kosaraju's algorithm in C, C++, Java and Python.</p>
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<span property="dc:title" content="Strongly Connected Components" class="rdf-meta element-hidden"></span>
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<p id="introduction">A strongly connected component is the portion of a directed graph in which there is a path from each vertex to another vertex. <strong>It is applicable only on</strong> <a href="/dsa/graph.html"><strong>a directed graph</strong></a>.</p>
<p>For example:</p>
<p>Let us take the graph below.</p>
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-initial-graph.png" title="initial graph" width="440" height="176"><figcaption>Initial graph</figcaption></figure><p>The strongly connected components of the above graph are:</p>
<figure><img alt="Strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-strongly-connected-components.png" title="Strongly connected components" width="476" height="230"><figcaption>Strongly connected components</figcaption></figure><p>You can observe that in the first strongly connected component, every vertex can reach the other vertex through the directed path.</p>
<p></p><div class="clearfix"></div><p>These components can be found using <strong>Kosaraju's Algorithm</strong>.</p>
<hr><h2 id="algorithm">Kosaraju's Algorithm</h2>
<p>Kosaraju's Algorithm is based on <a href="/dsa/graph-dfs.html">the depth-first search algorithm</a> implemented twice.</p>
<p>Three steps are involved.</p>
<ol><li>Perform a depth first search on the whole graph.<br><br>
Let us start from vertex-0, visit all of its child vertices, and mark the visited vertices as done. If a vertex leads to an already visited vertex, then push this vertex to the stack.<br><br>
For example: Starting from vertex-0, go to vertex-1, vertex-2, and then to vertex-3. Vertex-3 leads to already visited vertex-0, so push the source vertex (ie. vertex-3) into the stack.
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-step-1.png" title="step-1" width="451" height="304"><figcaption>DFS on the graph</figcaption></figure><br>
Go to the previous vertex (vertex-2) and visit its child vertices i.e. vertex-4, vertex-5, vertex-6 and vertex-7 sequentially. Since there is nowhere to go from vertex-7, push it into the stack.
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-step-2.png" title="step-2" width="451" height="304"><figcaption>DFS on the graph</figcaption></figure><br>
Go to the previous vertex (vertex-6) and visit its child vertices. But, all of its child vertices are visited, so push it into the stack.
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-step-3.png" title="step-3" width="451" height="152"><figcaption>Stacking</figcaption></figure><br>
Similarly, a final stack is created.
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-step-4.png" title="step-4" width="451" height="152"><figcaption>Final Stack</figcaption></figure></li>
<li>Reverse the original graph.
<figure><img alt="reversed graph" src="//cdn.programiz.com/sites/tutorial2program/files/scc-reversed-graph.png" title="reversed graph" width="440" height="176"><figcaption>DFS on reversed graph</figcaption></figure></li>
<li>Perform depth-first search on the reversed graph.<br><br>
Start from the top vertex of the stack. Traverse through all of its child vertices. Once the already visited vertex is reached, one strongly connected component is formed.<br><br>
For example: Pop vertex-0 from the stack. Starting from vertex-0, traverse through its child vertices (vertex-0, vertex-1, vertex-2, vertex-3 in sequence) and mark them as visited. The child of vertex-3 is already visited, so these visited vertices form one strongly connected component.
<figure><img alt="reversed graph - strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-reversed-step-1.png" title="reversed graph - step-1" width="465" height="417"><figcaption>Start from the top and traverse through all the vertices</figcaption></figure><br>
Go to the stack and pop the top vertex if already visited. Otherwise, choose the top vertex from the stack and traverse through its child vertices as presented above.
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/reversed%20step-2_0.png" title="step-2" width="451" height="404"><figcaption>Pop the top vertex if already visited</figcaption></figure>
<figure><img alt="reversed graph - strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/reversed%20step-3_0.png" title="reversed graph - step-3" width="451" height="389"><figcaption>Strongly connected component</figcaption></figure></li>
<li>Thus, the strongly connected components are:
<figure><img alt="strongly connected components" src="//cdn.programiz.com/sites/tutorial2program/files/scc-final-graph.png" title="strongly connected components" width="476" height="230"><figcaption>All strongly connected components</figcaption></figure></li>
</ol><hr><h2 id="code">Python, Java, C++ examples</h2>
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<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"><span class="hljs-comment"># Kosaraju's algorithm to find strongly connected components in Python</span>
<span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span>:</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">__init__</span><span class="hljs-params">(self, vertex)</span>:</span>
self.V = vertex
self.graph = defaultdict(list)
<span class="hljs-comment"># Add edge into the graph</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">add_edge</span><span class="hljs-params">(self, s, d)</span>:</span>
self.graph[s].append(d)
<span class="hljs-comment"># dfs</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">dfs</span><span class="hljs-params">(self, d, visited_vertex)</span>:</span>
visited_vertex[d] = <span class="hljs-literal">True</span>
<span class="hljs-keyword">print</span>(d, end=<span class="hljs-string">''</span>)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> self.graph[d]:
<span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> visited_vertex[i]:
self.dfs(i, visited_vertex)
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">fill_order</span><span class="hljs-params">(self, d, visited_vertex, stack)</span>:</span>
visited_vertex[d] = <span class="hljs-literal">True</span>
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> self.graph[d]:
<span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> visited_vertex[i]:
self.fill_order(i, visited_vertex, stack)
stack = stack.append(d)
<span class="hljs-comment"># transpose the matrix</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">transpose</span><span class="hljs-params">(self)</span>:</span>
g = Graph(self.V)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> self.graph:
<span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> self.graph[i]:
g.add_edge(j, i)
<span class="hljs-keyword">return</span> g
<span class="hljs-comment"># Print stongly connected components</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">print_scc</span><span class="hljs-params">(self)</span>:</span>
stack = []
visited_vertex = [<span class="hljs-literal">False</span>] * (self.V)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(self.V):
<span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> visited_vertex[i]:
self.fill_order(i, visited_vertex, stack)
gr = self.transpose()
visited_vertex = [<span class="hljs-literal">False</span>] * (self.V)
<span class="hljs-keyword">while</span> stack:
i = stack.pop()
<span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> visited_vertex[i]:
gr.dfs(i, visited_vertex)
<span class="hljs-keyword">print</span>(<span class="hljs-string">""</span>)
g = Graph(<span class="hljs-number">8</span>)
g.add_edge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>)
g.add_edge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>)
g.add_edge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>)
g.add_edge(<span class="hljs-number">2</span>, <span class="hljs-number">4</span>)
g.add_edge(<span class="hljs-number">3</span>, <span class="hljs-number">0</span>)
g.add_edge(<span class="hljs-number">4</span>, <span class="hljs-number">5</span>)
g.add_edge(<span class="hljs-number">5</span>, <span class="hljs-number">6</span>)
g.add_edge(<span class="hljs-number">6</span>, <span class="hljs-number">4</span>)
g.add_edge(<span class="hljs-number">6</span>, <span class="hljs-number">7</span>)
<span class="hljs-keyword">print</span>(<span class="hljs-string">"Strongly Connected Components:"</span>)
g.print_scc()</code></pre></div>
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<div class="code-editor__area" id="java-code">
<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs"><span class="hljs-comment">// Kosaraju's algorithm to find strongly connected components in Java</span>
<span class="hljs-keyword">import</span> java.util.*;
<span class="hljs-keyword">import</span> java.util.LinkedList;
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span> </span>{
<span class="hljs-keyword">private</span> <span class="hljs-keyword">int</span> V;
<span class="hljs-keyword">private</span> LinkedList<Integer> adj[];
<span class="hljs-comment">// Create a graph</span>
Graph(<span class="hljs-keyword">int</span> s) {
V = s;
adj = <span class="hljs-keyword">new</span> LinkedList[s];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < s; ++i)
adj[i] = <span class="hljs-keyword">new</span> LinkedList();
}
<span class="hljs-comment">// Add edge</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">int</span> d)</span> </span>{
adj[s].add(d);
}
<span class="hljs-comment">// DFS</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">DFSUtil</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">boolean</span> visitedVertices[])</span> </span>{
visitedVertices[s] = <span class="hljs-keyword">true</span>;
System.out.print(s + <span class="hljs-string">" "</span>);
<span class="hljs-keyword">int</span> n;
Iterator<Integer> i = adj[s].iterator();
<span class="hljs-keyword">while</span> (i.hasNext()) {
n = i.next();
<span class="hljs-keyword">if</span> (!visitedVertices[n])
DFSUtil(n, visitedVertices);
}
}
<span class="hljs-comment">// Transpose the graph</span>
<span class="hljs-function">Graph <span class="hljs-title">Transpose</span><span class="hljs-params">()</span> </span>{
Graph g = <span class="hljs-keyword">new</span> Graph(V);
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> s = <span class="hljs-number">0</span>; s < V; s++) {
Iterator<Integer> i = adj[s].listIterator();
<span class="hljs-keyword">while</span> (i.hasNext())
g.adj[i.next()].add(s);
}
<span class="hljs-keyword">return</span> g;
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">fillOrder</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">boolean</span> visitedVertices[], Stack stack)</span> </span>{
visitedVertices[s] = <span class="hljs-keyword">true</span>;
Iterator<Integer> i = adj[s].iterator();
<span class="hljs-keyword">while</span> (i.hasNext()) {
<span class="hljs-keyword">int</span> n = i.next();
<span class="hljs-keyword">if</span> (!visitedVertices[n])
fillOrder(n, visitedVertices, stack);
}
stack.push(<span class="hljs-keyword">new</span> Integer(s));
}
<span class="hljs-comment">// Print strongly connected component</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printSCC</span><span class="hljs-params">()</span> </span>{
Stack stack = <span class="hljs-keyword">new</span> Stack();
<span class="hljs-keyword">boolean</span> visitedVertices[] = <span class="hljs-keyword">new</span> <span class="hljs-keyword">boolean</span>[V];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
visitedVertices[i] = <span class="hljs-keyword">false</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
<span class="hljs-keyword">if</span> (visitedVertices[i] == <span class="hljs-keyword">false</span>)
fillOrder(i, visitedVertices, stack);
Graph gr = Transpose();
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
visitedVertices[i] = <span class="hljs-keyword">false</span>;
<span class="hljs-keyword">while</span> (stack.empty() == <span class="hljs-keyword">false</span>) {
<span class="hljs-keyword">int</span> s = (<span class="hljs-keyword">int</span>) stack.pop();
<span class="hljs-keyword">if</span> (visitedVertices[s] == <span class="hljs-keyword">false</span>) {
gr.DFSUtil(s, visitedVertices);
System.out.println();
}
}
}
<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">void</span> <span class="hljs-title">main</span><span class="hljs-params">(String args[])</span> </span>{
Graph g = <span class="hljs-keyword">new</span> Graph(<span class="hljs-number">8</span>);
g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>);
g.addEdge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>);
g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>);
g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">4</span>);
g.addEdge(<span class="hljs-number">3</span>, <span class="hljs-number">0</span>);
g.addEdge(<span class="hljs-number">4</span>, <span class="hljs-number">5</span>);
g.addEdge(<span class="hljs-number">5</span>, <span class="hljs-number">6</span>);
g.addEdge(<span class="hljs-number">6</span>, <span class="hljs-number">4</span>);
g.addEdge(<span class="hljs-number">6</span>, <span class="hljs-number">7</span>);
System.out.println(<span class="hljs-string">"Strongly Connected Components:"</span>);
g.printSCC();
}
}</code></pre></div>
</div>
<div class="code-editor__area" id="cpp-code">
<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs"><span class="hljs-comment">// Kosaraju's algorithm to find strongly connected components in C++</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><iostream></span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><list></span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stack></span></span>
<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> <span class="hljs-built_in">std</span>;
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span> {</span>
<span class="hljs-keyword">int</span> V;
<span class="hljs-built_in">list</span><<span class="hljs-keyword">int</span>> *adj;
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">fillOrder</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">bool</span> visitedV[], <span class="hljs-built_in">stack</span><<span class="hljs-keyword">int</span>> &Stack)</span></span>;
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">DFS</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">bool</span> visitedV[])</span></span>;
<span class="hljs-keyword">public</span>:
Graph(<span class="hljs-keyword">int</span> V);
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">int</span> d)</span></span>;
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printSCC</span><span class="hljs-params">()</span></span>;
<span class="hljs-function">Graph <span class="hljs-title">transpose</span><span class="hljs-params">()</span></span>;
};
Graph::Graph(<span class="hljs-keyword">int</span> V) {
<span class="hljs-keyword">this</span>->V = V;
adj = <span class="hljs-keyword">new</span> <span class="hljs-built_in">list</span><<span class="hljs-keyword">int</span>>[V];
}
<span class="hljs-comment">// DFS</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">Graph::DFS</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">bool</span> visitedV[])</span> </span>{
visitedV[s] = <span class="hljs-literal">true</span>;
<span class="hljs-built_in">cout</span> << s << <span class="hljs-string">" "</span>;
<span class="hljs-built_in">list</span><<span class="hljs-keyword">int</span>>::iterator i;
<span class="hljs-keyword">for</span> (i = adj[s].begin(); i != adj[s].end(); ++i)
<span class="hljs-keyword">if</span> (!visitedV[*i])
DFS(*i, visitedV);
}
<span class="hljs-comment">// Transpose</span>
<span class="hljs-function">Graph <span class="hljs-title">Graph::transpose</span><span class="hljs-params">()</span> </span>{
<span class="hljs-function">Graph <span class="hljs-title">g</span><span class="hljs-params">(V)</span></span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> s = <span class="hljs-number">0</span>; s < V; s++) {
<span class="hljs-built_in">list</span><<span class="hljs-keyword">int</span>>::iterator i;
<span class="hljs-keyword">for</span> (i = adj[s].begin(); i != adj[s].end(); ++i) {
g.adj[*i].push_back(s);
}
}
<span class="hljs-keyword">return</span> g;
}
<span class="hljs-comment">// Add edge into the graph</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">Graph::addEdge</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">int</span> d)</span> </span>{
adj[s].push_back(d);
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">Graph::fillOrder</span><span class="hljs-params">(<span class="hljs-keyword">int</span> s, <span class="hljs-keyword">bool</span> visitedV[], <span class="hljs-built_in">stack</span><<span class="hljs-keyword">int</span>> &Stack)</span> </span>{
visitedV[s] = <span class="hljs-literal">true</span>;
<span class="hljs-built_in">list</span><<span class="hljs-keyword">int</span>>::iterator i;
<span class="hljs-keyword">for</span> (i = adj[s].begin(); i != adj[s].end(); ++i)
<span class="hljs-keyword">if</span> (!visitedV[*i])
fillOrder(*i, visitedV, Stack);
Stack.push(s);
}
<span class="hljs-comment">// Print strongly connected component</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">Graph::printSCC</span><span class="hljs-params">()</span> </span>{
<span class="hljs-built_in">stack</span><<span class="hljs-keyword">int</span>> Stack;
<span class="hljs-keyword">bool</span> *visitedV = <span class="hljs-keyword">new</span> <span class="hljs-keyword">bool</span>[V];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
visitedV[i] = <span class="hljs-literal">false</span>;
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
<span class="hljs-keyword">if</span> (visitedV[i] == <span class="hljs-literal">false</span>)
fillOrder(i, visitedV, Stack);
Graph gr = transpose();
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
visitedV[i] = <span class="hljs-literal">false</span>;
<span class="hljs-keyword">while</span> (Stack.empty() == <span class="hljs-literal">false</span>) {
<span class="hljs-keyword">int</span> s = Stack.top();
Stack.pop();
<span class="hljs-keyword">if</span> (visitedV[s] == <span class="hljs-literal">false</span>) {
gr.DFS(s, visitedV);
<span class="hljs-built_in">cout</span> << <span class="hljs-built_in">endl</span>;
}
}
}
<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{
<span class="hljs-function">Graph <span class="hljs-title">g</span><span class="hljs-params">(<span class="hljs-number">8</span>)</span></span>;
g.addEdge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>);
g.addEdge(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>);
g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>);
g.addEdge(<span class="hljs-number">2</span>, <span class="hljs-number">4</span>);
g.addEdge(<span class="hljs-number">3</span>, <span class="hljs-number">0</span>);
g.addEdge(<span class="hljs-number">4</span>, <span class="hljs-number">5</span>);
g.addEdge(<span class="hljs-number">5</span>, <span class="hljs-number">6</span>);
g.addEdge(<span class="hljs-number">6</span>, <span class="hljs-number">4</span>);
g.addEdge(<span class="hljs-number">6</span>, <span class="hljs-number">7</span>);
<span class="hljs-built_in">cout</span> << <span class="hljs-string">"Strongly Connected Components:\n"</span>;
g.printSCC();
}</code></pre></div>
</div>
</div>
<hr><h2 id="complexity">Kosaraju's Algorithm Complexity</h2>
<p>Kosaraju's algorithm runs in linear time i.e. <code>O(V+E)</code>.</p>
<hr><h2 id="application">Strongly Connected Components Applications</h2>
<ul><li>Vehicle routing applications</li>
<li>Maps</li>
<li>Model-checking in formal verification</li>
</ul></div>
</div>
<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents
<button class="btn btn--clear align-items-center">
<svg class="programiz-icon"><use xlink:href="/sites/all/themes/programiz/assets/feather-sprite.svg#x"></use></svg></button></h3><div class="tutorial-toc__links"><ul><li><a href="#introduction">Introduction</a></li>
<li><a href="#algorithm">Kosaraju's Algorithm</a></li>
<li><a href="#code">Python, Java, C++ examples</a></li>
<li><a href="#complexity">Kosaraju's Algorithm Complexity</a></li>
<li><a href="#application">Strongly Connected Components Applications</a></li>
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