Graph Theory: Degree of regions in a planar graph

Question

-Background Information:

I am studying graph theory in discrete mathematics. As I was reading my notes, I came across an example provided by my professor that I am confused about. I need some clarification to understand it better, thanks.

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- Definition & Example:

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- Question:

If I start at a vertex and an edge and go all the way to around R2 and come back, should I pass the same edge and end the same vertex? (I assume each edge that connects two components is a bridge and a bridge counts as two edges).

Could you please see my thinking section down below, am I right with the way I solved the problem?

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- My thinking:

Consider each bridge to have two arrows indicating two edges.

In this case, I traverse 14 edges, so the degree of R4 is 14, am I right? Is this the way to do it?

If yes, why not using the other (green) edge, using that edge can give us a shorter closed walk of 12? (considering you go and come back from that edge).

Answer 1

You need to count each edge bordering the region:

Answer 2

Yes, there are $6+4$ "normal" edges bordering $R4$ and $2$ bridges, giving $10+2\cdot 2=14$.