jmc

We must find the length of each side of the hexagon to find the perimeter.

We can see that the distance between each pair of points $(1,2)$ and $(2,2)$, $(2,2)$ and $(2,1)$, and $(2,1)$ and $(3,1)$ is 1. Thus, these three sides have a total length of 3.

We can see that the distance between $(0,1)$ and $(1,2)$ is $\sqrt{2}$. The distance between $(3,1)$ and $(2,0)$ is also $\sqrt{2}$. These two sides have a total length of $2\sqrt{2}$.

We can see that the distance between $(2,0)$ and $(0,1)$ is $\sqrt{5}$. Thus, the last side has length of $\sqrt{5}$.

Summing all of these distances, we find that the perimeter is $3+2\sqrt{2}+1\sqrt{5}$, so $a+b+c=\boxed{6}$.