Slitherlink Signatures

Let $G$ be a planar graph and let $C$ be a cycle in $G$. Inside of each finite face of $G$, we write down the number of edges of that face which belong to $C$. This is the signature of $C$ in $G$. The notion of a signature arises naturally in the context of Slitherlink puzzles. The signature of a cycle does not always determine it uniquely. We focus on the ambiguity of signatures in the case when $G$ is a rectangular grid of unit square cells. We describe all grids which admit an ambiguous signature. For each such grid, we then determine the greatest possible difference between two cycles with the same signature on it. We also study the possible values of the total number of cycles which fit a given signature. We discuss various related questions as well.