Edited: I realize know I was misled by the non-graph theoretical) use of the word edge (where the question title strongly suggested a graph algorithm question :))
Why do you suppose there are no edges? There are many logical discrete locations that you could model, and limited connections between (i.e. not where a wall is :)). There you go: you have your edges.
What you probably mean is that you don't want to represent your edges in data (which you don't have to, but still there are the logical edges that connect locations/points.)
That said:
you ask whether someone has seen things like this before. I vaguely recall seeing something relevant to this in Knuths Dancing Links
article (DLX) which is an implementation technique for A* algorithms.
The article specifically treats states as 'cells' (in a grid) with east/west/north/south links. It's been a long time so I don't quite recall how you would map (no pun intended) your problem on that algorithm.
The dance steps. One good way to implement algorithm X is to represent each 1 in the
matrix A as a data object x with five fields L[x]; R[x]; U [x]; D[x]; C[x]. Rows of the matrix
are doubly linked as circular lists via the L and R fields ("left" and "right"); columns are
doubly linked as circular lists via the U and D fields ("up" and "down"). Each column
list also includes a special data object called its list header.