Complete walk around the boundary of faces in a planar graph For a planar graph $(G)$ and a face $(f)$ of $G$, the degree of $f$ denoted deg$(f)$, is the number of (possibly non-distinct) edges traversed in a complete walk around the boundary of $f$.
I have trouble understanding the statement because I have never heard the term complete walk in graph theory nor I could find any examples, an example would be appreciated.

Basically it is just "how many sides does the face have" -- except that an edge will count as "two sides" if it has the same face on both sides of it, such as the BX edge here:
A-----------E
|           |
| 1 Y---Z   |
|   | 2 |   |
B---X---W   |     3
|           |
C-----------D

A "complete walk around the face" marked 1 on the drawing would be
A-B-X-Y-Z-W-X-B-C-D-E-A 

which traverses BX in both directions.
I don't think "complete" is really a technical term here; it's just an attempt to express the idea that you're walking all the way around the face, keeping it outer wall on your right (as above) or left the entire way.
