Playing along a single string is quite un-guitaristic, but can give new ideas on phrasing, as it does not rely on usual vertical/diagonal patterns and allows playing wide intervals easily with hammer-ons and pull-offs.

All examples will be shown with the open string as root note; the study can be transposed to any string.

Let's begin with a major scale:

Simply moving or taking away notes we can form other scales.

The mixolydian scale has a minor 7th: 

Here's the major pentatonic:

Now the natural minor:

From it the dorian scale:

Harmonic minor:

Minor pentatonic:

Now a symmetrical, exotic one (Whole-tone):

Now let's apply this concept to 7th arpeggios.

Major 7th:

Minor 7th:

Dominant 7th:

Half-diminished (m7, b5):

Diminished:

This is just a starting point, and the idea can be applied to any scale or arpeggio.