Aussie Maths & Science Teachers: Save your time with SmarterEd
The network below can be represented as a planar graph.
Redraw the graph as a planar representation of the network, labelling each vertex. (2 marks)
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`text{Redrawing the graph in planar form (no edges crossing):}`
The network below can be represented as a planar graph.
Redraw the graph as a planar representation of the network, labelling each vertex. (2 marks)
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`text{Redrawing the graph in planar form (no edges crossing):}`
The graph above has
`text(Redrawing the graph in planar form,)`
`text(the graph can be seen to have 6 faces.)`
`text(Alternatively, using Euler’s rule:)`
| `v + f` | `= e + 2` |
| `5 + f` | `= 9 + 2` |
| `:. f` | `= 6` |
`=> B`
The network below can be represented as a planar graph.
Redraw the graph as a planar representation of the network, labelling each vertex. (2 marks)
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`text{Redrawing the graph in planar form (no edges crossing):}`
The network below can be represented as a planar graph.
Complete the partial graph drawn below, adding the missing edges so that it is a planar representation of the above network. (3 marks)
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`text{Redrawing the graph in planar form (no edges crossing):}`
The network below can be represented as a planar graph.
Draw the planar graph representation of this network, labelling each vertex. (2 marks)
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`text{Redrawing the graph in planar form (no edges crossing):}`
`text{Each vertex is the same degree as the original graph and}`
`text{has edges connecting to the same vertices.}`
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(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` |
A connected planar graph has 4 edges and 4 faces.
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i. `text{2 vertices}`
ii.
| i. `v+f` | `=e+2` |
| `:. v` | `=e-f + 2` |
| `= 4-4 + 2` | |
| `= 2` |
ii.
Consider the graph below.
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i.
ii. \(\text{Number of faces = 4}\)
i.
ii. \(\text{Method 1}\)
\(\text{By inspection, number of faces = 4}\)
\(\text{Method 2}\)
| `v-e+f` | `=2` | |
| `4-6+f` | `=2` | |
| `f` | `=4` |
A network is represented by the following graph.
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i.
ii. \(6\)
i.
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` |
The graph below represents a friendship network. The vertices represent the four people in the friendship network: Kwan (K), Louise (L), Milly (M) and Narelle (N).
An edge represents the presence of a friendship between a pair of these people. For example, the edge connecting K and L shows that Kwan and Louise are friends.
Which one of the following graphs does not contain the same information?
