Question

the following graph is planar. (trust me.) if it is drawn with no intersecting edges, into how many regions (\faces\) does it cut up the plane?
2
3
faces =
1
4
8
5
7
6
submit question

Explanation:

Step1: Count Vertices (V)

Vertices: 1,2,3,4,5,6,7,8 → \( V = 8 \)

Step2: Count Edges (E)

Count each edge:
- 1-8, 2-8, 2-3, 2-4, 2-7, 3-4, 3-5, 3-7, 4-5, 4-6, 4-7, 5-6, 5-7, 6-7, 8-5, 8-6, 8-7 (Wait, better way: list all connections. Let's recount carefully:
From 1: 1-8 (1)
From 2: 2-8, 2-3, 2-4, 2-7 (4) → total 1+4=5
From 3: 3-4, 3-5, 3-7 (3) → total 5+3=8
From 4: 4-5, 4-6, 4-7 (3) → total 8+3=11
From 5: 5-6, 5-7 (2) → total 11+2=13
From 6: 6-7 (1) → total 13+1=14
From 8: 8-5, 8-6, 8-7 (3) → Wait, earlier missed? Wait no, 2-8,1-8, then 8-5,8-6,8-7. Wait, let's list all edges:
1-8, 2-8, 2-3, 2-4, 2-7, 3-4, 3-5, 3-7, 4-5, 4-6, 4-7, 5-6, 5-7, 6-7, 8-5, 8-6, 8-7. Wait, that's 17? Wait no, maybe better to use hand count:
Looking at the graph:
- 1 connected to 8 (1)
- 2 connected to 8,3,4,7 (4) → 1+4=5
- 3 connected to 2,4,5,7 (4? Wait 3-2, 3-4, 3-5, 3-7 → 4) → 5+4=9
- 4 connected to 2,3,5,6,7 (5? 4-2,4-3,4-5,4-6,4-7 → 5) → 9+5=14
- 5 connected to 3,4,6,7,8 (5? 5-3,5-4,5-6,5-7,5-8 → 5) → 14+5=19
- 6 connected to 4,5,7,8 (4? 6-4,6-5,6-7,6-8 → 4) → 19+4=23? No, this is wrong. Let's use Euler's formula: \( F = E - V + 2 \), but we need correct E.
Alternative: Let's count edges properly:
Nodes: 8. Let's list adjacency:
1: [8] → 1 edge
2: [8,3,4,7] → 4 edges (total 1+4=5)
3: [2,4,5,7] → 4 edges (5+4=9)
4: [2,3,5,6,7] → 5 edges (9+5=14)
5: [3,4,6,7,8] → 5 edges (14+5=19)
6: [4,5,7,8] → 4 edges (19+4=23)
7: [2,3,4,5,6,8] → 6 edges (23+6=29? No, this is overcounting because each edge is shared. Wait, no: each edge is between two nodes, so we should count each edge once. Let's use a better approach:
List all unique edges:
1-8,
2-8, 2-3, 2-4, 2-7,
3-4, 3-5, 3-7,
4-5, 4-6, 4-7,
5-6, 5-7, 5-8,
6-7, 6-8,
7-8 (Wait, 7-8? Wait node 7 and 8: is there an edge? In the graph, 8 is connected to 1,2,5,6,7? Let's see the graph: 8 is connected to 1 (vertical), 2 (line), 5 (line), 6 (line), 7 (line). So 8-1, 8-2, 8-5, 8-6, 8-7 → 5 edges from 8.
2 is connected to 8,3,4,7 → 4 edges.
3 is connected to 2,4,5,7 → 4 edges.
4 is connected to 2,3,5,6,7 → 5 edges.
5 is connected to 3,4,6,7,8 → 5 edges.
6 is connected to 4,5,7,8 → 4 edges.
7 is connected to 2,3,4,5,6,8 → 6 edges.
1 is connected to 8 → 1 edge.
Now, sum all edges (each edge counted twice, once from each end), so total edges \( E = \frac{1 + 4 + 4 + 5 + 5 + 4 + 6}{2} \)? Wait no: 1 (from 1) + 4 (2) +4 (3)+5(4)+5(5)+4(6)+6(7)+5(8) → Wait 8 nodes: 1,2,3,4,5,6,7,8. Their degrees:
1:1, 2:4, 3:4, 4:5, 5:5, 6:4, 7:6, 8:5.
Sum of degrees: 1+4+4+5+5+4+6+5 = 34. So \( E = \frac{34}{2} = 17 \). Yes, because each edge contributes to two degrees. So \( E = 17 \).

Step3: Apply Euler's Formula for Planar Graphs

Euler's formula: \( V - E + F = 2 \) (for connected planar graphs; check if connected: all nodes are connected through 8,2, etc. So connected.)
We have \( V = 8 \), \( E = 17 \).
Solve for \( F \):
\( F = E - V + 2 \)
\( F = 17 - 8 + 2 = 11 \)? Wait no, wait Euler's formula is \( V - E + F = 2 \) → \( F = E - V + 2 \). Wait 17-8=9, 9+2=11? But wait, maybe I miscounted E. Let's recount degrees:
Node 1: degree 1 (correct, only 1-8)
Node 2: connected to 8,3,4,7 → degree 4 (correct)
Node 3: connected to 2,4,5,7 → degree 4 (correct)
Node 4: connected to 2,3,5,6,7 → degree 5 (correct)
Node 5: connected to 3,4,6,7,8 → degree 5 (correct)
Node 6: connected to 4,5,7,8 → degree 4 (correct)
Node 7: connected to 2,3,4,5,6,8 → degree 6 (correct)
Node 8: connected to 1,2,5,6,7 → degree 5 (correct)
Sum: 1+4=5, +4=9, +5=14, +5=19, +4=23, +6=29, +5=34. So \( E = 34/2 = 17 \). Correct.
Then \( F = 17 - 8 + 2 = 11 \)? Wait but let's check with another approach. Let's list the faces. Wait maybe I made a mistake in E. Wait let's count edges manually:
Edges:
1-8,
2-8, 2-3, 2-4, 2-7,
3-4, 3-5, 3-7,
4-5, 4-6, 4-7,
5-6, 5-7, 5-8,
6-7, 6-8,
7-8 (Wait, 7-8: is that an edge? In the graph, 7 and 8: yes, as per degree of 8 (connected to 7) and degree of 7 (connected to 8). So that's an edge. Wait, but in the initial count, when I listed edges, did I miss 7-8? Let's list all 17 edges:
1. 1-8
2. 2-8
3. 2-3
4. 2-4
5. 2-7
6. 3-4
7. 3-5
8. 3-7
9. 4-5
10. 4-6
11. 4-7
12. 5-6
13. 5-7
14. 5-8
15. 6-7
16. 6-8
17. 7-8
Yes, that's 17 edges. Now, Euler's formula: \( V=8 \), \( E=17 \), so \( F = E - V + 2 = 17 - 8 + 2 = 11 \). Wait but let's check with a simple graph. Wait maybe the graph is not connected? Wait node 1 is connected to 8, which is connected to others, so yes, connected. So Euler's formula applies. Thus, the number of faces (regions) is 11. Wait, but let's verify with another method. Let's count the outer face and inner faces. But Euler's formula is reliable for connected planar graphs. So I think the calculation is correct.

Answer:

11