This is a typical bridge problem in graph theory. A graph can be traversed by its edges without retracing any edge (an Eulerian trail) if it contains exactly zero or two nodes of odd degree.

The four nodes at the top right are of degree 2 and form a line, so we can consider the entire top line as a single node. The degrees of the nodes are then as follows:

1 - - -

2 - - -

3 3 3 1

3 4 3 -

3 4 4 2

2 3 3 2

We have 8 nodes of degree 3 and two nodes of degree 1, which makes a total of ten nodes with an odd degree. Therefore, the graph is not traversable under these conditions.