Yup, that's the idea, although it's operating per-channel. So you'll get the min red, min green, min blue and min alpha in your output. Not sure if that's what you want. If you are looking for overall pixel brightness you may want to get the luminance of the pixel
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dot(color, vec3(0.3, 0.6, 0.1))
and do a minimum of that. There is no general 'correct' way to do this, it totally depends on what you need.
The boundary pixel values you get can be controlled by the Extend Mode menu in the GLSL TOP.