# Understanding surface code diagrams

I'm trying to understand the diagrams used in the following paper. I'm new to the area so I don't have experience with the stabilizer formalism, but I'm trying to figure out what's going on in the diagrams (Figures 1 and 2, specifically). I have a few specific questions:

1. Each vertex has a physical data qubit. But what exactly do the operators (green circles) represent? Is there an X and Z operator on every vertex site, or only some of them (Figure 1b)?

2. Relatedly, the logical X and Z operators are specified as chains of single-qubit operators (the green circles, I presume) connecting top to bottom (logical X) or left to right (logical Z). Do these chains have to be straight lines, or can they twist and turn?

3. Do the semi-circular plaquettes (which have only two single-qubit operators on them) represent the fact that we are wrapping around the surface (so we are on a torus), or is it something else?

4. I don't understand the single-qubit Pauli flips described in Figure 2. I can't see how we get an error (red dot). Any explanation for this would be helpful.

If someone could also explain to me the relationship between the Figure 1 in the above paper and Figure 1 of this paper, I think it would help makes things clearer.