…that the Grabengasse in the old town runs along the defence ditch from medieval times?

The oldest part of the city wall is called the Römerwehr (Roman Defence Wall) and it still exists to be found arching around the Seminary garden on Cathedral Square. Unfortunately, the garden is out of bounds of tourists!
Between the Danube and the Inn on the high rock of Cathedral Hill the defence ditch runs between the old and new part of the city. The Grabengasse was built in front of this impressive 2 to 3 m. broad and 6 to 8 m. high wall. This part of the defence fortification is not Roman as one is led to believe by the name.
First built at the end of the 10^th century, it received its name at the end of the 19^th century thanks to the mode of the time,  the love of Roman and Greek antiquity.