If your maze is a maze without islands (that is, walls which are not connected to any wall surrounding the maze) and each corridor is at most one cell wide, there is a really nice way to reduce the maze to the path to the cheese, if such a path exists :

• Look at the maze as an array of cells.
• Every cell that is not a wall is considered alive.
• Every cell that is a wall is considered dead.
• A cell that has three dead neighbours dies as well.
• A cell that has none,one or two dead neighbours lives.
• Stop when in one turn no cell died.

When this algorithm stops, you end up either with the path from the mouse to the cheese, or with an enclosed mouse (or cheese).