Supppse we wanted to write a solver for this kind of puzzle. This could be considered a type of integer programming problem, where each square is indexed by a variable. So if the puzzle is $N \times N$ there are $N^2$ variables and a number of restrictions.
Somehow we could encode the restriction there's a single path connecting the dots on each vertex. And that every vertex must be covered.
Is this equivalent to one of the more standard CS problems such as set-cover or SAT?
Related: Are "Flow Free" puzzles NP-hard?