Generate OPEN surface mesh from a set of 3D points

I have a set of points on an OPEN surface in 3D space. I have identified a subset of points which lay on the boundary. I mean to generate a triangulation of those points, which gives me an open surface and keeps my selected points on the boundary.

All references I found deal with (sometimes?) closed surfaces, e.g., CGAL. See examples below. In addition, some CGAL algorithms require oriented normals at each point, which I do not have.

Is there an available algorithm and code for this? (either CGAL Advancing_front_surface_reconstruction, properly handled, or any other)

See also this and this.

Example 1

I compiled and ran example reconstruction_surface_mesh.cpp from examples/Advancing_front_surface_reconstruction, out-of-the box (which uses file half.xyz as input for data points), and I obtained a closed surface:

I would like to get rid of the few triangles that close the surface. I tried adding an extra point at the end of half.xyz, and I got

which is an open surface. So far, with what I tested, I do not know:

How to indicate an open surface.
How to indicate which vertices lay at the boundary. If this is a non-empty set (and it should have at least three vertices) this would imply an open surface.

Ideally, one would have a workflow which works without manual intervention.

Example 2

I compiled and ran example boundaries.cpp, out-of-the box (which also uses file half.xyz as input for data points). The output is:

0 outliers:

Boundaries:
boundary
0.178269 0.438589 0.129521
0.0795598 0.419465 0.244812
0.0549683 0.377617 0.3119
-0.0295721 0.360972 0.329075
-0.111332 0.334417 0.342617
-0.186667 0.2953 0.346683
-0.2719 0.16555 0.375017
-0.336304 0.117058 0.339323
-0.393517 0.0775 0.285917
-0.421419 -0.126854 0.215271
-0.395217 -0.214417 0.20015
-0.354783 -0.2953 0.170767
-0.237067 -0.395867 0.172233
-0.178246 -0.438588 0.129553
0.0227767 -0.4873 0.0700833
0.220338 -0.438589 -7.23321e-06
0.293 -0.395867 0
0.36025 -0.334417 0
0.418077 -0.258382 6.0303e-05
0.46025 -0.17265 0
0.484417 -0.0425167 -0.0763333
0.485067 0.03875 -0.0782667
0.471547 0.117058 -0.076827
0.44605 0.197567 -0.0700833
0.4092 0.27125 -0.0433167
0.364885 0.329645 0
0.313633 0.377617 0.0441167
0.2509 0.41425 0.0879333

I did not find how to use this for automatically removing triangles which would make my target boundary vertices not laying at the boundary.

Moreover, the output seems to be the list of boundary points, without the "spurious" triangles (I am not sure). I would like to specify this list.

The CGAL advancing front reconstruction algorithm does generate open surfaces in general.