smc-4789-35-Circuit

A network diagram is drawn below.

Starting at vertex `Z`, identify a trail that uses 6 edges and ends at vertex `V`. (1 mark)

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Starting at vertex `V`, identify all six different paths that end at vertex `Y`. (2 marks)

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Is the route identified as `YXUVXY` a circuit? Explain. (1 mark)

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i. `text{Trail: repeated vertices permitted but no repeated edges}`

`text{Trail (6 edges)}\ Z–V:`

`ZWXYUXV\ \ \text{or}\ \ ZWXUYXV`

ii. `text{Path: a route that does not repeat vertices or edges.}`

`text{Possible paths}\ V–Y:`

`VUY, VUXY, VUXWZY`

`VXUY, VXY, VXWZY`

iii. `text{Circuit: starts and ends at the same vertex with repeated}`

`text{vertices allowed but no repeated edges.}`

`YXUVXY\ \text{is not a circuit because it edge}\ XY\ \text{is repeated.}`

Show Worked Solution

i. `text{Trail: repeated vertices permitted but no repeated edges}`

`text{Trail (6 edges)}\ Z–V:`

`ZWXYUXV\ \ \text{or}\ \ ZWXUYXV`

ii. `text{Path: a route that does not repeat vertices or edges.}`

`text{Possible paths}\ V–Y:`

`VUY, VUXY, VUXWZY`

`VXUY, VXY, VXWZY`

iii. `text{Circuit: starts and ends at the same vertex with repeated}`

`text{vertices allowed but no repeated edges.}`

`YXUVXY\ \text{is not a circuit because it edge}\ XY\ \text{is repeated.}`

Starting at vertex `A`, identify all the different paths that finish at vertex `D`, using only three edges. (2 marks)

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Starting at vertex `A`, identify a cycle route? (1 mark)

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Explain the difference between a cycle and a circuit route. (1 mark)

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i. `text{Three edge paths from}\ A–D:`

`ACED, ABCD, ABED, AECD`

ii. `text{Cycle: a path that ends at the same vertex without}`

`text{repeating any.}`

`text{Possible cycles (many examples exist in this network):}`

`ABEA, ABCEA, ACBA, …`

iii. `text{A circuit is a path that begins and ends at the same vertex.}`

`text{A cycle begins and ends at the same vertex, without}`

`text{repeating any vertices.}`