The Quadrangulate Node is designed to remove obvious edges from a triangulated mesh in order to return to what the original quaded mesh would've looked like.
It works as a 2 step process, the first step is identifying the diagonal edges in the quads, this can be done iteratively or by assuming a good vertex order.
Those edges are then deleted, but in some cases this also deletes polygons that were tris on the original mesh.
The second step works by fixing those edges that have more than 4 points, you can either use a polyFill aproach to fix those areas, or another iterative approach where it divides the mesh up until quads are left.
Delete every Nth edge of a model
Iteratively remove the longest edge in the mesh until there are only quads left
Edge to Collapse
While in Vertex Order Mode this selects which edge of the triangle to collapse, 1st, 2nd or 3rd
Avoid deletion of edges that break the silhouette
When Protect Silhouette is on, this determines the angle between 2 faces to break in.
Resolve NGons by using the polyFill Quadrilaterals option.
Iterative slice the NGons until they are all quads or tris.
Fallback Fill Mode
When in PolyFill mode, how to resolve cases where the Quadrilaterals option fails
Edge to Dissolve
When in Iterative Divide mode, how to determine which edge to be divided