You can draw an open GraphicsPath
with a thin or a thick Pen. But a region
must be set from a closed shape or else there is no place where your pixels could show up. This will help to keep your region intact; but you need to know, just what you want it to be:
if (shape != ShapeType.Line) this.Region = new Region(path);
If you want it to be something like a thick line you must create a polygon or a series of lines to outline the shape you want. And if you want your line to be inside that region you will need two paths: one closed polygon path to set the region and one open line path to draw the line inside the region.
Edit:
The best way to create the closed path is probably to use the Widen()
method with the Pen you are using like this:
GraphicsPath path2 = path.Widen(yourPen);
This would get the thickness right as well as the line caps and also work for more complicated polylines; I haven't tried it though..