Cycle Detection in Directed Graph in DSA Go - 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.

intermediate

Which data structure is commonly used to detect cycles in a directed graph using DFS?

A recursion stack (or call stack) is used along with a visited set to detect back edges indicating cycles.

intermediate

What does a back edge represent in DFS traversal of a directed graph?

A back edge points from a node to one of its ancestors in the DFS tree, indicating a cycle.

advanced

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

Because a node can be visited multiple times in different paths; we need to track nodes in the current recursion stack to detect cycles.

beginner

What is the time complexity of cycle detection in a directed graph using DFS?

The time complexity is O(V + E), where V is vertices and E is edges, since each node and edge is visited once.

What indicates a cycle during DFS in a directed graph?

A Finding a back edge to a node in the current recursion stack

B Visiting a node for the first time

C Finding a node with no outgoing edges

D Completing DFS without revisiting nodes

A back edge to a node in the current recursion stack means a cycle exists.

Which of these is NOT needed for cycle detection in a directed graph using DFS?

A Visited set

B Queue for BFS

C Recursion stack

D Graph adjacency list

Queue for BFS is not used in DFS-based cycle detection.

What does the recursion stack track in cycle detection?

A All visited nodes

B Nodes with no incoming edges

C Nodes currently being explored in the DFS path

D Nodes already processed

The recursion stack tracks nodes in the current DFS path to detect cycles.

If no back edge is found during DFS, what can we conclude?

A The graph is undirected

B The graph is disconnected

C The graph has at least one cycle

D The graph has no cycles

No back edge means no cycle in the directed graph.

What is the main difference between cycle detection in directed vs undirected graphs?

A Directed graphs require tracking recursion stack; undirected graphs do not

B Undirected graphs use recursion stack; directed graphs do not

C Cycle detection is impossible in directed graphs

D Both use the same method

Directed graphs need recursion stack tracking to detect cycles; undirected graphs use parent tracking.

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

Think about tracking nodes currently in the path and revisiting them.

Describe why a recursion stack is necessary for cycle detection in directed graphs but not in undirected graphs.

Consider how cycles appear differently in directed vs undirected graphs.

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 the direction of edges

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 edge directions.

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 the direction of edges -> Option A

4. Quick Check:

   Cycle detection = finding directed loops [OK]

Hint: Cycle means a loop following edge directions [OK]

Common Mistakes:

• Confusing cycle with shortest path
• Ignoring edge directions
• Thinking cycle means any repeated node

2. Which of the following Go code snippets correctly declares a graph as an adjacency list for cycle detection?

easy

A. graph := make(map[int][]int)

B. graph := []int{}

C. graph := map[int]int{}

D. graph := [][]int{}

Solution

1. Step 1: Understand adjacency list representation

   An adjacency list for a directed graph maps each node to a list of nodes it points to.

2. Step 2: Check Go syntax for adjacency list

   Using map[int][]int creates a map from int to slice of ints, perfect for adjacency list.

3. Final Answer:

   graph := make(map[int][]int) -> Option A

4. Quick Check:

   Adjacency list = map[int][]int [OK]

Hint: Adjacency list uses map of int to slice of int [OK]

Common Mistakes:

• Using slice of int only (no mapping)
• Using map[int]int which stores single int per node
• Using [][]int without map keys

3. What will be the output of the following Go code snippet detecting cycle in a directed graph?

package main
import "fmt"

func dfs(node int, graph map[int][]int, visited, recStack map[int]bool) bool {
    if recStack[node] {
        return true
    }
    if visited[node] {
        return false
    }
    visited[node] = true
    recStack[node] = true
    for _, neighbor := range graph[node] {
        if dfs(neighbor, graph, visited, recStack) {
            return true
        }
    }
    recStack[node] = false
    return false
}

func main() {
    graph := map[int][]int{
        1: {2},
        2: {3},
        3: {1},
    }
    visited := make(map[int]bool)
    recStack := make(map[int]bool)
    fmt.Println(dfs(1, graph, visited, recStack))
}

medium

A. false

B. true

C. Compilation error

D. Runtime panic

Solution

1. Step 1: Analyze graph structure

   Graph edges: 1->2, 2->3, 3->1 form a cycle.

2. Step 2: Trace DFS cycle detection

   DFS from node 1 visits 2, then 3, then back to 1 which is in recursion stack, so returns true.

3. Final Answer:

   true -> Option B

4. Quick Check:

   Cycle detected = true [OK]

Hint: Cycle if node revisited in recursion stack [OK]

Common Mistakes:

• Ignoring recursion stack for cycle detection
• Assuming visited alone detects cycle
• Confusing false with no cycle

4. Identify the bug in the following Go code snippet for cycle detection in a directed graph:

func dfs(node int, graph map[int][]int, visited map[int]bool) bool {
    if visited[node] {
        return true
    }
    visited[node] = true
    for _, neighbor := range graph[node] {
        if dfs(neighbor, graph, visited) {
            return true
        }
    }
    return false
}

medium

A. Return type should be int, not bool

B. Visited map should be a slice, not a map

C. Missing recursion stack to track nodes in current path

D. Graph should be a slice of slices, not a map

Solution

1. Step 1: Understand cycle detection logic

   Cycle detection in directed graph needs to track nodes in current recursion path (recursion stack).

2. Step 2: Identify missing recursion stack

   This code only uses visited to mark visited nodes, which cannot detect cycles correctly in directed graphs.

3. Final Answer:

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

4. Quick Check:

   Recursion stack needed for directed cycle detection [OK]

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

Common Mistakes:

• Using only visited to detect cycles
• Confusing undirected and directed cycle detection
• Ignoring recursion stack leads to false negatives

5. Given a directed graph represented as adjacency list:

graph := map[int][]int{
    1: {2, 3},
    2: {4},
    3: {4, 5},
    4: {6},
    5: {6},
    6: {3},
}

Which nodes are part of a cycle in this graph?

hard

A. [1, 3, 5]

B. [1, 2, 5]

C. [2, 4, 5]

D. [3, 4, 6]

Solution

1. Step 1: Analyze edges for cycles

   Edges: 3->4, 4->6, 6->3 form a cycle involving nodes 3, 4, 6.

2. Step 2: Confirm cycle nodes

   Nodes 1, 2, 5 lead into cycle but are not part of cycle themselves.

3. Final Answer:

   [3, 4, 6] -> Option D

4. Quick Check:

   Cycle nodes = [3,4,6] [OK]

Hint: Trace back edges to find cycle nodes [OK]

Common Mistakes:

• Including nodes that only lead to cycle
• Missing nodes that form the cycle
• Confusing cycle membership with reachability