This document discusses a map of the Danube river flowing through Budapest, Hungary showing bridges connecting areas of land. It introduces representing the system of rivers and bridges as a graph, where areas of land are vertices and bridges are edges. The graph drawn on the map shows the connections between vertices without regard for their actual positions. Changes to the graph, such as adding or removing edges, could impact whether a route crossing each bridge once is possible.

1. Budapest, Hungary
Rivers and Bridges
This is a map of the Danube flowing through the city of Budapest in Hungary
Can you draw a path that goes over each of the bridges once and only once?
Do you think it is possible?
Can you prove this?
Do you think adding bridges onto either of the stretches of river that go off
the page would change whether finding such a path was possible? Why?
We can simplify this problem into one that we can look at mathematically by
drawing the system of rivers and bridges as a graph. By a graph we mean a
diagram that shows all the important connections. We mark each area of
land as a point (called a vertex) and each bridge as a line (called an edge)
joining the two points that represent the land that bridge joins.
Have a go at drawing this on top of the map and turn over to see if you got it
right.

2. Here is what the graph looks like on top of the map.
Because we have all the important information, we can move the vertices
and edges around as much as we like and the graph is the same so long as
we don’t add or remove edges or vertices.
Try drawing your own version of this graph. Can you make it look more or
less complicated while keeping it the same?
Do you think you could change
whether a route is possible just
by adding or removing one
edge (bridge)?
Can you spot any requirements
for a path to be possible?