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Why do this problem?

Introductions to graph theory can often end up being quite dry and dusty, with lots of definitions that need to be memorised. This problem invites students to engage with the different types of graph as a pictorial representation so that they can understand why different categorisations are necessary.

Possible approach

Hand out the cards.
"Here is a set of cards. Each card shows a representation of a graph. A graph is a collection of vertices, also called nodes, joined by edges, also called arcs. Sort the cards into groups that you think belong together in some way, and sketch the graphs together with an explanation of why you have sorted them together. See how many different ways of categorising the graphs you can come up with."

While students are working, circulate and look out for groups who have come up with collections of graphs that fit standard definitions. Here are some definitions that might be appropriate.

Bring the class together and share the categorisations they came up with. When they identify a standard categorisation, share the usual terminology with them. Perhaps hand out the definitions sheet and invite them to find examples on the cards for each definition.

Key questions

What is it about some graphs that make them similar to others?
Can you think of different properties of the graphs?
Can you trace routes from one node to another?

Possible extension

This article discusses properties of graphs and leads to a proof of Euler's formula.

Possible support

All students should be able to make a start on this activity by sorting the cards and explaining why they have chosen to group certain graphs together.