Families of m-convex polygons: m=1

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their 'concavity index', m. Such polygons are called m-convex polygons. We first use the inclusion-exclusion principle to rederive the known generating function for 1-convex self-avoiding polygons (SAPs). We then use our results to derive the exact anisotropic generating functions for osculating and neighbour-avoiding 1-convex SAPs, their isotropic form having recently been conjectured.