Connectivity

Expert Used by Smart Hint

The finished loop is a single connected shape, so every line already drawn has to end up joined to every other one. An edge whose removal would cut one group of lines off from another is forced in; an edge marooned in a pocket the drawn lines can never reach is ruled out.

Edge on the loop
Edge ruled out
Cells that establish the pattern

How it works

The finished loop is ONE closed shape. So every line already on the board has to end up joined to every other one — and the loop can only ever travel along edges you haven't ruled out.
That turns the X's into walls. Ask where the loop can still physically go, and whole areas can drop out at once.
Here, follow the two edges at the top-left corner dot. The top one leads to the next dot along, whose other two edges are both X'd. The left one leads down to a dot in exactly the same state. Those two edges are sealed into a dead pocket.
A line already exists down in the bottom-right, so the loop runs through there — and it can't be in two disconnected places at once. Both pocket edges are ruled out.
The same argument runs the other way too, and it's the more useful half: if taking an edge AWAY would cut one group of drawn lines off from another, the loop has no other route between them, so that edge is forced in. Look for narrow necks between two separate runs of line.