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7. Maximum Independent Set Cardinality (Numerical)

Adjacency matrix:

$$ A = \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 0 \ 1 & 0 & 1 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 & 1 & 1 \ 1 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 & 1 & 0 \end{bmatrix} $$

Solution:

Let the vertices be 1-6. Let’s try to find the largest independent set (set of vertices no two of which are adjacent):

  • Vertex 1: adjacent to 2, 4.
  • Vertex 2: adjacent to 1, 3, 4.
  • Vertex 3: adjacent to 2, 5, 6.
  • Vertex 4: adjacent to 1, 2.
  • Vertex 5: adjacent to 3, 6.
  • Vertex 6: adjacent to 3, 5.

Try picking 1, 5:

  • 1 is connected to 2, 4.
  • 5 is connected to 3, 6 (neither are 1 or 5). Try 1, 5, 4: 1 and 4 connected (invalid).

Try 1, 3, 6: 1 and 3 are not connected, 1 and 6 not connected, 3 and 6 are connected (invalid).

Try 1, 5, 3: 3 and 5 connected (invalid).

Try 4, 5, 6: 4 and 5 not connected, 4 and 6 not connected, 5 and 6 are connected (invalid).

Try 1, 5:

  • 1 (not adjacent to 5).
  • 5 (adjacent to 3 and 6).
  • Add 4? 1 and 4 connected (invalid).
  • Add 6? 5 and 6 connected (invalid).
  • Add 2? 1 and 2 connected (invalid).

Try 1, 5 — that’s 2. Try 1, 6: 1 and 6 not connected.

  • 1 (adjacent to 2, 4)
  • 6 (adjacent to 3,5)
  • Add 4? 1 and 4 connected (invalid).

Try 2, 5, 4:

  • 2 and 5 not connected.
  • 2 and 4 connected (invalid).

Try 4, 5:

  • 4 and 5 not connected.

Try 4, 6:

  • 4 and 6 not connected.

Try 4, 5, 2: 4 and 2 connected.

Try 1, 5, 6: 5 and 6 connected.

Try 1, 5.

Try 4, 6.

Try 2, 5.

Try 2 and 6.

No independent set of size 3.

But what about 3,4,5?

  • 3 and 4 not connected.
  • 3 and 5 connected.

So, maximum independent set is of size 2.

Answer: 2

Summary Table

Qn Answer/Explanation
1 $ n(n-1) $
2 (b) $ 5,5,2,2,1,1 $; (c) $ 5,3,3,2,2,1 $
3 Stack
4 (8,11)
5 Second graph
6 (a), (b)
7 2
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