diff --git a/.gitignore b/.gitignore
index ac796da..7fb050f 100644
--- a/.gitignore
+++ b/.gitignore
@@ -24,6 +24,3 @@ x86/
 *.vcxproj
 *.vcxproj.user
 *.vcxproj.filters
-
-*.pdf
-*.pptx
\ No newline at end of file
diff --git a/algorithm/tarjan.h b/algorithm/tarjan.h
new file mode 100644
index 0000000..3a72da6
--- /dev/null
+++ b/algorithm/tarjan.h
@@ -0,0 +1,74 @@
+#ifndef TARJAN_H_
+#define TARJAN_H_
+
+#include "graph/graph.h"
+using namespace graph;
+
+#include <stack>
+#include <iostream>
+#include <ranges>
+#include <iterator>
+#include <algorithm>
+
+namespace algorithm {
+
+template<typename T>
+class Tarjan {
+public:
+  Tarjan(Graph<T> G) : G(G) {}
+
+  void findSCC();
+
+  void strongConnect(Vertex<T>& v);
+
+  std::uint16_t index = 0;
+  std::stack<Vertex<T>> S;
+private:
+  Graph<T> G;
+};
+
+template<typename T>
+void Tarjan<T>::strongConnect(Vertex<T>& v) {
+  v.index, v.lowlink = index++;
+  v.onStack = true;
+  S.push(v);
+
+  for (auto& w : v.edges) {
+    if (w.index == -1) {
+      // successor w has not yet been visited, recurse on it
+      strongConnect(w);
+      v.lowlink = std::min(v.lowlink, w.lowlink);
+    } else if (w.onStack) {
+      // successor w is in stack S and hence in the current SCC
+      // if w is not on stack, then (v, w) is an edge pointing to an SCC
+      // already found and must be ignored
+      v.lowlink = std::min(v.lowlink, w.index);
+    }
+  }
+
+  // if v is a root node, pop the stack and generate an SCC
+  if (v.lowlink = v.index) {
+    // start a new SCC
+    Vertex<T> w;
+    do {
+      w = S.top();
+      S.pop();
+      // add w to current SCC
+
+    } while (w != v);
+  }
+}
+
+template<typename T>
+void Tarjan<T>::findSCC() {
+
+  for (auto& vertex : G.vertices) {
+    if (vertex.index == -1) {
+      strongConnect(vertex);
+    }
+  }
+}
+
+} // namespace algorithm
+
+#endif
\ No newline at end of file
diff --git a/doc/IMPROVED DYNAMIC REACHABILITY ALGORITHMS.pdf b/doc/IMPROVED DYNAMIC REACHABILITY ALGORITHMS.pdf
new file mode 100644
index 0000000..797cbdd
Binary files /dev/null and b/doc/IMPROVED DYNAMIC REACHABILITY ALGORITHMS.pdf differ
diff --git a/graph/graph.h b/graph/graph.h
new file mode 100644
index 0000000..a661953
--- /dev/null
+++ b/graph/graph.h
@@ -0,0 +1,65 @@
+#ifndef GRAPH_H_
+#define GRAPH_H_
+
+#include "vertex.h"
+
+#include <map>
+#include <set>
+#include <vector>
+#include <utility>
+
+namespace graph {
+
+template<typename T>
+class Graph {
+public:
+  Graph() = default;
+
+  // Add vertex v
+  void insert(Vertex<T>& v);
+
+  // Add edge between v and u
+  void insert(Vertex<T> v, Vertex<T> u);
+
+  // Delete vertex v
+  void erase(Vertex<T>& v);
+  
+  // Delete edge between v and u
+  void erase(Vertex<T>& v, Vertex<T>& u);
+
+  // Return true if v and u are connected
+  bool connected(const Vertex<T>& v, const Vertex<T>& u) const;
+
+  // Adjacency matrix representation
+  std::vector<Vertex<T>> vertices;
+};
+
+template<typename T>
+inline void Graph<T>::insert(Vertex<T>& v) {
+  vertices.push_back(v);
+}
+
+template<typename T>
+inline void Graph<T>::insert(Vertex<T> v, Vertex<T> u) {
+  v.edges.push_back(u);
+  vertices.push_back(v);
+}
+
+template<typename T>
+inline void Graph<T>::erase(Vertex<T>& v) {
+  //
+}
+
+template<typename T>
+inline void Graph<T>::erase(Vertex<T>& v, Vertex<T>& u) {
+  //
+}
+
+template<typename T>
+inline bool Graph<T>::connected(const Vertex<T>& v, const Vertex<T>& u) const {
+  return false;
+}
+
+} // namespace graph
+
+#endif
\ No newline at end of file
diff --git a/graph/vertex.h b/graph/vertex.h
new file mode 100644
index 0000000..9d447ee
--- /dev/null
+++ b/graph/vertex.h
@@ -0,0 +1,36 @@
+#ifndef VERTEX_H_
+#define VERTEX_H_
+
+#include <cstdint>
+#include <compare>
+#include <ostream>
+#include <vector>
+
+namespace graph {
+
+template<typename T>
+class Vertex {
+public:
+  Vertex() = default;
+  Vertex(T v) : v(v) {}
+  
+  T v;
+  std::uint16_t index = -1;
+  std::uint16_t lowlink = -1;
+  bool onStack = false;
+  std::vector<Vertex<T>> edges;
+
+  auto operator<=>(const Vertex&) const = default;
+
+  template<typename T>
+  friend inline std::ostream& operator<<(std::ostream& os, const Vertex<T>& v);
+};
+
+template<typename T>
+inline std::ostream& operator<<(std::ostream& os, const Vertex<T>& vout) {
+  return os << vout.v;
+}
+
+} // namespace graph
+
+#endif
\ No newline at end of file
diff --git a/include/graph.h b/include/graph.h
deleted file mode 100644
index a90bd60..0000000
--- a/include/graph.h
+++ /dev/null
@@ -1,61 +0,0 @@
-#ifndef GRAPH_H_
-#define GRAPH_H_
-
-#include <map>
-#include <set>
-#include <vector>
-#include <utility>
-
-namespace graph {
-
-template<typename V>
-class Graph {
-public:
-  Graph() = default;
-
-  // Add vertex v
-  void insert(const V& v);
-
-  // Add edge between v and u
-  void insert(const V& v, const V& u);
-
-  // Delete vertex v
-  void erase(const V& v);
-  
-  // Delete edge between v and u
-  void erase(const V& v, const V& u);
-
-  // Return true if v and u are connected
-  bool connected(const V& v, const V& u);
-private:
-  std::map<V, std::set<V>> adjacency;
-};
-
-template<typename V>
-inline void Graph<V>::insert(const V& v) {
-  adjacency[v];
-}
-
-template<typename V>
-inline void Graph<V>::insert(const V& v, const V& u) {
-  adjacency[v].insert(u);
-}
-
-template<typename V>
-inline void Graph<V>::erase(const V& v) {
-  adjacency.erase(v);
-}
-
-template<typename V>
-inline void Graph<V>::erase(const V& v, const V& u) {
-  adjacency[v].erase(u);
-}
-
-template<typename V>
-inline bool Graph<V>::connected(const V& v, const V& u) {
-  return true;  
-}
-
-} // namespace graph
-
-#endif
\ No newline at end of file
diff --git a/test/graph_test.cc b/test/graph_test.cc
index 7ec5bb4..7720b65 100644
--- a/test/graph_test.cc
+++ b/test/graph_test.cc
@@ -1,11 +1,13 @@
 #include <doctest/doctest.h>
 
-#include <cstdint>
+#include "graph/graph.h"
+using namespace graph;
+#include "algorithm/tarjan.h"
 
-#include "include/graph.h"
+#include <cstdint>
+#include <iostream>
 
 TEST_SUITE("Testing Graph.") {
-  using namespace graph;
 
   TEST_CASE("Insert vertices/edges.") {
     Graph<std::uint16_t> G;
@@ -19,6 +21,12 @@ TEST_SUITE("Testing Graph.") {
     G.insert(3, 6);
     G.insert(4, 6);
     G.insert(5, 6);
+    
+    Vertex<std::uint16_t> v1(1);
+    Vertex<std::uint16_t> v2(3);
+    Vertex<std::uint16_t> v3(5);
+    algorithm::Tarjan<std::uint16_t> tarjan(G);
+    tarjan.findSCC();
 
     // CHECK_EQ(G.connected(1, 5), true);