Bipartite Graph Check in DSA C - Execution Trace

Concept Flow - Bipartite Graph Check

Start with graph

↓

Pick uncolored node

↓

Assign color 0

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Use BFS/DFS to visit neighbors

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If neighbor uncolored -> assign opposite color

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If neighbor colored same as current -> Not Bipartite

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Repeat for all nodes

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If no conflicts -> Graph is Bipartite

We start with an uncolored graph, pick nodes one by one, color them and their neighbors with opposite colors using BFS or DFS. If a conflict arises, graph is not bipartite.

Execution Sample

DSA C

void bfsCheck(int start) {
  queue<int> q;
  color[start] = 0;
  q.push(start);
  while (!q.empty()) {
    int u = q.front(); q.pop();
    for (int v : adj[u]) {
      if (color[v] == -1) {
        color[v] = 1 - color[u];
        q.push(v);
      } else if (color[v] == color[u]) {
        bipartite = false;
      }
    }
  }
}

This BFS colors nodes starting from 'start'. If a neighbor has the same color, it marks graph as not bipartite.

Execution Table

Step	Operation	Node Visited	Neighbor	Color Assigned	Conflict Detected	Color Array State	Visual State
1	Start BFS at node 0	0	-	color[0]=0	No	[0:0,1:-1,2:-1,3:-1]	0(0) 1(-) 2(-) 3(-)
2	Visit neighbors of 0	0	1	color[1]=1	No	[0:0,1:1,2:-1,3:-1]	0(0) 1(1) 2(-) 3(-)
3	Visit neighbors of 0	0	3	color[3]=1	No	[0:0,1:1,2:-1,3:1]	0(0) 1(1) 2(-) 3(1)
4	Visit node 1	1	0	-	No	[0:0,1:1,2:-1,3:1]	0(0) 1(1) 2(-) 3(1)
5	Visit neighbors of 1	1	2	color[2]=0	No	[0:0,1:1,2:0,3:1]	0(0) 1(1) 2(0) 3(1)
6	Visit node 3	3	0	-	No	[0:0,1:1,2:0,3:1]	0(0) 1(1) 2(0) 3(1)
7	Visit neighbors of 3	3	2	-	No	[0:0,1:1,2:0,3:1]	0(0) 1(1) 2(0) 3(1)
8	Stop BFS	-	-	-	No	[0:0,1:1,2:0,3:1]	No conflict found, graph is bipartite

💡 No conflict detected; graph is bipartite.

Variable Tracker

Variable	Start	After Step 1	After Step 3	After Step 5	After Step 7	Final
color	[-1,-1,-1,-1]	[0,-1,-1,-1]	[0,1,-1,1]	[0,1,0,1]	[0,1,0,1]	[0,1,0,1]
bipartite	true	true	true	true	true	true
queue	[]	[0]	[1,3]	[3,2]	[3,2]	[]

Key Moments - 3 Insights

Why do we assign opposite colors to neighbors?

Assigning opposite colors ensures no two connected nodes share the same color, which is the definition of bipartite. See execution_table steps 2 and 5 where neighbors get opposite colors.

What causes the conflict detection to mark graph as not bipartite?

When a neighbor already colored has the same color as the current node, it means the graph cannot be split into two sets properly. This happens at step 7 in execution_table.

Why do we start BFS from uncolored nodes?

Starting BFS from uncolored nodes ensures all disconnected parts of the graph are checked. The first step colors node 0 to start the process.

Visual Quiz - 3 Questions

Look at the execution_table, what color is assigned to node 3 at step 3?

A -1 (uncolored)

B 0

C 1

D Conflict detected

💡 Hint

Check the 'Color Assigned' and 'Color Array State' columns at step 3.

At which step does the conflict get detected?

A Step 7

B Step 4

C Step 5

D Step 2

💡 Hint

Look at the 'Conflict Detected' column in execution_table.

If node 2 was colored 0 instead of 1 at step 5, what would happen at step 7?

A Conflict detected earlier

B No conflict detected

C Queue would be empty

D Graph would be bipartite

💡 Hint

Refer to color assignments and conflict logic in execution_table steps 5 and 7.

Concept Snapshot

Bipartite Graph Check:
- Use BFS or DFS to color nodes with 0 or 1
- Assign opposite color to neighbors
- If neighbor has same color, graph is not bipartite
- Check all connected components
- Return true if no conflicts found

Full Transcript

To check if a graph is bipartite, we color nodes using two colors (0 and 1). We start from an uncolored node, assign color 0, then use BFS to assign opposite colors to neighbors. If at any point a neighbor has the same color as the current node, the graph is not bipartite. We repeat this for all nodes to cover disconnected parts. The execution table shows step-by-step how nodes are colored and when conflict is detected. The variable tracker shows color array changes and queue state. Key moments clarify why opposite colors are assigned and how conflicts arise. The quiz tests understanding of color assignments and conflict detection steps.