Bipartite Graph Check in DSA Rust - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a bipartite graph?

A bipartite graph is a graph whose nodes can be divided into two groups so that no edge connects nodes within the same group.

beginner

How can you check if a graph is bipartite using colors?

You try to color each node with one of two colors so that no two connected nodes share the same color. If this is possible, the graph is bipartite.

intermediate

What data structure is commonly used to perform the bipartite check?

A queue is often used in a Breadth-First Search (BFS) to traverse nodes and assign colors while checking for conflicts.

beginner

What does it mean if during the bipartite check you find two connected nodes with the same color?

It means the graph is not bipartite because nodes in the same group are connected, which breaks the bipartite rule.

intermediate

Why do we need to check all nodes in the graph when checking bipartiteness?

Because the graph might be disconnected, so we must check each connected part to ensure the entire graph is bipartite.

What is the main idea behind checking if a graph is bipartite?

A Color nodes with two colors so no connected nodes share the same color

B Count the number of edges

C Check if the graph is fully connected

D Find the shortest path between nodes

The bipartite check colors nodes with two colors ensuring no two connected nodes have the same color.

Which graph traversal method is commonly used for bipartite checking?

A Depth-First Search (DFS)

B Breadth-First Search (BFS)

C Dijkstra's Algorithm

D Prim's Algorithm

BFS is commonly used to assign colors level by level during bipartite checking.

If two connected nodes have the same color during the check, what does it mean?

A The graph is complete

B The graph is bipartite

C The graph is disconnected

D The graph is not bipartite

Same color on connected nodes breaks bipartite rules, so the graph is not bipartite.

Why must all nodes be checked in a bipartite graph check?

A Because the graph might be disconnected

B To calculate degrees

C To count edges

D To find the shortest path

Disconnected parts might have different bipartite properties, so all nodes must be checked.

What colors are used in bipartite graph checking?

A Three colors

B Only one color

C Two colors

D Four colors

Two colors are used to separate nodes into two groups.

Explain how to check if a graph is bipartite using BFS and coloring.

Think about coloring nodes level by level and checking neighbors.

Describe what a bipartite graph is and why it matters in graph problems.

Imagine two teams where no one plays with their own team.

Practice

1. What does it mean for a graph to be bipartite?

easy

A. Its vertices can be divided into two groups with no edges inside the same group

B. It has an equal number of vertices and edges

C. It contains at least one cycle of odd length

D. All vertices have the same degree

Solution

1. Step 1: Understand bipartite graph definition

   A bipartite graph is one where vertices can be split into two sets so that no edge connects vertices within the same set.

2. Step 2: Check options against definition

   Its vertices can be divided into two groups with no edges inside the same group matches this definition exactly. Options A, B, and D describe unrelated properties.

3. Final Answer:

   Its vertices can be divided into two groups with no edges inside the same group -> Option A

4. Quick Check:

   Bipartite graph = two groups, no internal edges [OK]

Hint: Bipartite means two groups with no edges inside groups [OK]

Common Mistakes:

• Confusing bipartite with equal vertices and edges
• Thinking bipartite graphs must have cycles
• Assuming all vertices have same degree

2. Which Rust code snippet correctly declares an adjacency list for an undirected graph to check bipartiteness?

easy

A. let graph: Vec<(usize, usize)> = Vec::new();

B. let graph: Vec<usize> = vec![0; n];

C. let graph: HashMap = HashMap::new();

D. let graph: Vec = vec![vec![]; n];

Solution

1. Step 1: Identify adjacency list type

   An adjacency list for a graph is usually a vector of vectors, where each inner vector holds neighbors of a vertex.

2. Step 2: Match code snippet to adjacency list

   let graph: Vec = vec![vec![]; n]; declares a vector of vectors of usize, initialized with empty vectors for each vertex, which is correct. Options A, B, and C do not represent adjacency lists properly.

3. Final Answer:

   let graph: Vec = vec![vec![]; n]; -> Option D

4. Quick Check:

   Adjacency list = Vec of Vecs [OK]

Hint: Adjacency list is Vec in Rust [OK]

Common Mistakes:

• Using Vec<usize> instead of Vec
• Using HashMap with wrong types
• Using tuple vector which is edge list, not adjacency list

3. Given the following Rust code snippet to check bipartiteness, what is the output?

let graph = vec![
    vec![1, 3],
    vec![0, 2],
    vec![1, 3],
    vec![0, 2],
];

fn is_bipartite(graph: &Vec) -> bool {
    let n = graph.len();
    let mut colors = vec![-1; n];
    for i in 0..n {
        if colors[i] == -1 {
            let mut stack = vec![i];
            colors[i] = 0;
            while let Some(u) = stack.pop() {
                for &v in &graph[u] {
                    if colors[v] == -1 {
                        colors[v] = 1 - colors[u];
                        stack.push(v);
                    } else if colors[v] == colors[u] {
                        return false;
                    }
                }
            }
        }
    }
    true
}

println!("{}", is_bipartite(&graph));

medium

A. true

B. false

C. Compilation error

D. Runtime panic

Solution

1. Step 1: Analyze graph structure

   The graph has 4 vertices with edges forming a square (0-1-2-3-0). This is an even cycle, which is bipartite.

2. Step 2: Trace DFS coloring

   The code colors vertex 0 as 0, then neighbors 1 and 3 as 1, then neighbors 2 as 0. No conflicts occur, so it returns true.

3. Final Answer:

   true -> Option A

4. Quick Check:

   Even cycle graph is bipartite = true [OK]

Hint: Even cycles are bipartite, so output is true [OK]

Common Mistakes:

• Thinking all cycles are non-bipartite
• Confusing stack usage with queue
• Missing color conflict check

4. The following Rust code attempts to check if a graph is bipartite but has a bug. What is the bug?

fn is_bipartite(graph: &Vec) -> bool {
    let n = graph.len();
    let mut colors = vec![-1; n];
    for i in 0..n {
        if colors[i] == -1 {
            let mut stack = vec![i];
            colors[i] = 0;
            while let Some(u) = stack.pop() {
                for &v in &graph[u] {
                    if colors[v] == -1 {
                        colors[v] = colors[u]; // Bug here
                        stack.push(v);
                    } else if colors[v] == colors[u] {
                        return false;
                    }
                }
            }
        }
    }
    true
}

medium

A. Stack is used instead of queue, causing infinite loop

B. Colors assigned to neighbors are same as current vertex, causing wrong bipartite check

C. colors vector is initialized with zeros instead of -1

D. Graph parameter should be mutable

Solution

1. Step 1: Identify color assignment bug

   The code assigns neighbors the same color as current vertex, which breaks bipartite property requiring opposite colors.

2. Step 2: Understand impact on bipartite check

   This causes false negatives or positives because neighbors must have opposite colors, so the check fails.

3. Final Answer:

   Colors assigned to neighbors are same as current vertex, causing wrong bipartite check -> Option B

4. Quick Check:

   Neighbors must have opposite colors [OK]

Hint: Neighbors must get opposite color, not same [OK]

Common Mistakes:

• Assigning same color to neighbors
• Confusing stack and queue usage
• Initializing colors incorrectly

5. You have a graph with vertices 0 to 5 and edges: (0-1), (1-2), (2-3), (3-0), (3-4), (4-5). Which of the following statements is true about its bipartiteness?

hard

A. The graph is not bipartite because it contains an odd cycle (3-4-5)

B. The graph is not bipartite because it contains an even cycle (0-1-2-3)

C. The graph is bipartite because it contains no odd cycles

D. The graph is bipartite because all vertices have degree 2 or less

Solution

1. Step 1: Identify cycles in the graph

   The graph has a cycle 0-1-2-3-0 which is length 4 (even). The path 3-4-5 is a chain, not a cycle.

2. Step 2: Check bipartite condition

   Graphs with no odd cycles are bipartite. Since the only cycle is even, the graph is bipartite.

3. Final Answer:

   The graph is bipartite because it contains no odd cycles -> Option C

4. Quick Check:

   No odd cycles means bipartite [OK]

Hint: Check for odd cycles; none means bipartite [OK]

Common Mistakes:

• Mistaking chains for cycles
• Thinking even cycles break bipartiteness
• Assuming degree affects bipartiteness