Cycle Detection in Directed Graph in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a cycle in a directed graph?

A cycle in a directed graph is a path that starts and ends at the same vertex, following the direction of edges.

beginner

Which data structures are commonly used to detect cycles in a directed graph?

Depth-First Search (DFS) with recursion stack or colors (white, gray, black) is commonly used to detect cycles in directed graphs.

intermediate

What does the recursion stack represent in DFS cycle detection?

The recursion stack keeps track of vertices currently in the path of DFS. If a vertex is revisited while still in the recursion stack, a cycle exists.

intermediate

Explain the color method for cycle detection in directed graphs.

Vertices are colored white (unvisited), gray (visited but not finished), or black (finished). Encountering a gray vertex during DFS means a cycle is detected.

intermediate

Why can't we use simple visited arrays alone to detect cycles in directed graphs?

Because revisiting a vertex that is already visited but not in the current DFS path does not mean a cycle. We need to track the current path (recursion stack or colors) to detect cycles.

What indicates a cycle during DFS in a directed graph?

A Visiting a vertex with only one incoming edge

B Visiting a vertex that is currently in the recursion stack

C Visiting a vertex with no outgoing edges

D Visiting a vertex that was visited in a previous DFS call

A cycle is detected if DFS visits a vertex that is still in the recursion stack (current path).

Which color represents a vertex that is currently being explored in the color method?

A Gray

B White

C Black

D Blue

Gray means the vertex is visited but DFS has not finished exploring all its descendants.

What is the time complexity of cycle detection using DFS in a directed graph with V vertices and E edges?

A O(V + E)

B O(V^2)

C O(E^2)

D O(V * E)

DFS visits each vertex and edge once, so the time complexity is O(V + E).

If no cycle is found in a directed graph, what does that imply about the graph?

A The graph is a tree

B The graph is disconnected

C The graph has no edges

D The graph is acyclic

No cycle means the graph is acyclic, also called a Directed Acyclic Graph (DAG).

Which of the following is NOT a valid method to detect cycles in a directed graph?

A DFS with recursion stack

B Color method with white, gray, black

C Breadth-First Search (BFS) with queue only

D Topological sort failure detection

BFS alone cannot detect cycles in directed graphs without additional mechanisms.

Describe how DFS can be used to detect a cycle in a directed graph.

Think about how to know if a vertex is revisited in the current path.

Explain the difference between visited array and recursion stack in cycle detection.

Focus on why just knowing if a vertex was visited before is not enough.

Practice

1. What does cycle detection in a directed graph help us find?

easy

A. Whether there is a path that starts and ends at the same node following edge directions

B. The shortest path between two nodes

C. The total number of nodes in the graph

D. The maximum degree of any node

Solution

1. Step 1: Understand cycle in directed graph

   A cycle means a path that starts and ends at the same node following the direction of edges.

2. Step 2: Identify what cycle detection finds

   Cycle detection checks if such a path exists in the graph.

3. Final Answer:

   Whether there is a path that starts and ends at the same node following edge directions -> Option A

4. Quick Check:

   Cycle detection = path starting and ending at same node [OK]

Hint: Cycle means path loops back to start following edges [OK]

Common Mistakes:

• Confusing cycle with shortest path
• Thinking cycle ignores edge directions
• Mixing cycle detection with counting nodes

2. Which of the following is the correct way to declare a visited array for cycle detection in C for a graph with n nodes?

easy

A. int visited = new int[n];

B. int visited[n] = {0};

C. int visited[] = {0};

D. int visited[n];

Solution

1. Step 1: Understand array declaration in C

   In C, to declare an array of size n, use int visited[n]; but it is uninitialized.

2. Step 2: Initialize array to zero

   To initialize all elements to zero, use int visited[n] = {0}; which sets all to zero.

3. Final Answer:

   int visited[n] = {0}; -> Option B

4. Quick Check:

   Array declaration with zero init = int visited[n] = {0}; [OK]

Hint: Use int arr[n] = {0} to initialize all zeros in C [OK]

Common Mistakes:

• Using 'new' keyword which is C++ not C
• Declaring array without size
• Incorrect initialization syntax

3. Consider the following C code snippet for cycle detection in a directed graph using DFS. What will be the output?

void dfs(int node, int visited[], int recStack[], int graph[][3]) {
    visited[node] = 1;
    recStack[node] = 1;
    for (int i = 0; i < 3; i++) {
        int adj = graph[node][i];
        if (adj == -1) continue;
        if (!visited[adj]) dfs(adj, visited, recStack, graph);
        else if (recStack[adj]) printf("Cycle detected\n");
    }
    recStack[node] = 0;
}

int main() {
    int graph[3][3] = {{1, -1, -1}, {2, -1, -1}, {0, -1, -1}};
    int visited[3] = {0}, recStack[3] = {0};
    dfs(0, visited, recStack, graph);
    return 0;
}

medium

A. No output

B. Infinite loop

C. Segmentation fault

D. Cycle detected

Solution

1. Step 1: Analyze graph edges

   Graph edges: 0->1, 1->2, 2->0 form a cycle.

2. Step 2: Trace DFS calls and cycle detection

   DFS visits 0, then 1, then 2. When 2 tries to visit 0 again which is in recursion stack, it prints "Cycle detected".

3. Final Answer:

   Cycle detected -> Option D

4. Quick Check:

   Cycle in graph triggers print "Cycle detected" [OK]

Hint: Cycle prints when node revisited in recursion stack [OK]

Common Mistakes:

• Ignoring recursion stack for cycle detection
• Assuming no output means no cycle
• Confusing visited and recursion stack arrays

4. The following code attempts to detect a cycle in a directed graph but does not work correctly. What is the main error?

void dfs(int node, int visited[], int graph[][3]) {
    visited[node] = 1;
    for (int i = 0; i < 3; i++) {
        int adj = graph[node][i];
        if (adj == -1) continue;
        if (!visited[adj]) dfs(adj, visited, graph);
        else printf("Cycle detected\n");
    }
}

int main() {
    int graph[3][3] = {{1, -1, -1}, {2, -1, -1}, {0, -1, -1}};
    int visited[3] = {0};
    dfs(0, visited, graph);
    return 0;
}

medium

A. Graph adjacency matrix is incorrect

B. Visited array is not initialized

C. Missing recursion stack to track nodes in current path

D. DFS function missing return type

Solution

1. Step 1: Understand cycle detection logic

   Cycle detection in directed graph needs to track nodes in current recursion path using a recursion stack.

2. Step 2: Identify missing recursion stack

   This code only uses visited array, so it cannot distinguish back edges causing cycles from cross edges.

3. Final Answer:

   Missing recursion stack to track nodes in current path -> Option C

4. Quick Check:

   Recursion stack needed for correct cycle detection [OK]

Hint: Use recursion stack to detect back edges in DFS [OK]

Common Mistakes:

• Assuming visited array alone detects cycles
• Ignoring need to track current DFS path
• Confusing undirected and directed cycle detection

5. You have a directed graph represented as adjacency lists:

0: [1]
1: [2]
2: [3, 4]
3: [1]
4: []

Which nodes are part of a cycle detected by DFS cycle detection algorithm?

hard

A. Nodes 1, 2, 3

B. Nodes 0, 1, 4

C. Nodes 0, 2, 4

D. Nodes 3, 4

Solution

1. Step 1: Trace the graph edges

   Edges: 0->1, 1->2, 2->3 and 2->4, 3->1.

2. Step 2: Identify cycle nodes

   Cycle is 1->2->3->1. Nodes 1, 2, 3 form the cycle.

3. Final Answer:

   Nodes 1, 2, 3 -> Option A

4. Quick Check:

   Cycle nodes = 1, 2, 3 [OK]

Hint: Follow edges to find loops including nodes [OK]

Common Mistakes:

• Including nodes not in cycle like 0 or 4
• Missing node 3 in cycle
• Confusing cycle with path