smc-4788-50-Euler’s formula

Networks, SMB-014

The network below can be represented as a planar graph.

Draw the planar graph representation of this network and find the number of faces in the planar graph. (3 marks)

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`text{Number of faces = 6}`

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`text(Redrawing the graph in planar form:)`

`text{Method 1}`

`text{Number of faces = 6 (by inspection)}`

`text(Method 2 (Euler’s formula))`

`v + f`	`= e + 2`
`5 + f`	`= 9 + 2`
`:. f`	`= 6`

Networks, SMB-013

Consider the planar graph below.

Euler’s formula will be verified for this graph.

Find the values of `e, v` and `f` and use them to verify Euler's formula. (3 marks)

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`e=6, v=5, f=3`

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`text{Number of edges:}\ e=6`

`text{Number of vertices:}\ v=5`

`text{Number of faces:}\ f=3`

`text(Verifying Euler’s formula):\ \ v + f = e + 2`

`5+3`	`= 6+2`
`8`	`= 8\ \ =>\ \text{Euler holds}`

Networks, SMB-012

A connected planar graph has 4 edges and 4 faces.

Calculate the number of vertices for this graph. (2 marks)
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Draw the planar graph. (2 marks)
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i. `text{2 vertices}`

ii.

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i. `v+f`	`=e+2`
`:. v`	`=e-f + 2`
	`= 4-4 + 2`
	`= 2`

ii.

Networks, SMB-011 MC

A connected planar graph has 10 edges and 10 faces.

The number of vertices for this graph is

`2`
`5`
`8`
`12`

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`A`

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`v+f`	`=e+2`
`:. v`	`= e-f + 2`
	`= 10-10 + 2`
	`= 2`

`=> A`

Networks, SMB-010 MC

A connected planar graph has seven vertices and nine edges.

The number of faces that this graph will have is

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`D`

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`v + f `	`= e + 2`
`7 + f `	`= 11`
`f`	`= 4`

`=> D`

Networks, SMB-009 MC

A planar graph has five faces.

This graph could have

six vertices and eight edges.
eight vertices and five edges.
eight vertices and six edges.
five vertices and eight edges.

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`D`

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`text(Using Euler’s formula:)`

`v + f`	`= e + 2`
`v + 5`	`= e + 2`
`v + 3`	`= e`

`text{Consider each option:}`

`text{5 vertices and 8 edges → Euler holds}`

`=> D`

Networks, SMB-008

A planar graph has five vertices and six faces.

Calculate the number of edges in the graph. (2 marks)

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`9`

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`v + f`	`= e + 2`
`5 + 6`	`= e + 2`
`:. e`	`= 9`

Networks, SMB-007 MC

A connected planar graph has 12 edges.

This graph could have

5 vertices and 6 faces.
5 vertices and 8 faces.
6 vertices and 8 faces.
6 vertices and 9 faces.

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`C`

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`text(Consider option C:)`

`v + f`	`= e + 2`
`6 + 8`	`= 12 + 2`
`14`	`= 14`

`text(i.e. Euler’s formula holds.)`

`=> C`

Networks, SMB-006

Consider the graph below.

Redraw this network as a planar graph. (1 mark)

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Find the number of faces on the planar graph. (2 marks)

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ii. \(\text{Number of faces = 4}\)

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ii. \(\text{Method 1}\)

\(\text{By inspection, number of faces = 4}\)

\(\text{Method 2}\)

`v-e+f`	`=2`
`4-6+f`	`=2`
`f`	`=4`

Networks, SMB-005

A network is represented by the following graph.

Draw the above network as a planar graph. (1 mark)

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Find the number of faces of this planar graph. (2 marks)
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ii. \(6\)

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ii. \(\text{Method 1}\)

\(\text{By inspection (see image above):} \)

\(\text{Number of faces = 6} \)

\(\text{Method 2}\)

`v-e+f`	`=2`
`8-12+f`	`=2`
`f`	`=6`