Self-avoiding closed curves in the regular and semiregular grids

We consider closed curves in the three regular and eight semiregular grids in the plane, in which each vertex and each edge can be repeated a limited number of times. We define the conditions for such curves to be self-avoiding, and we present a linear-time algorithm to check them. We define the orientation of such curves. We propose a classification of their vertices, and we give a unifying formula relating the number of different types of vertices, valid in the regular and semiregular grids. Our results can be used in the plane tiling applications.