|these weeds have scalloped green leaves & seem to grow on runners...if you start to pull some out, you get a string of plants...they're taking over my mulched beds around the edges,(are thick in|
|Based on your description I think it is probably ground ivy (known by many different names) or Glechoma hederacea. There is a photo of it here -- you may need to cut and paste the complete url into your browser to make it work correctly.
This weed is very difficult to control. In my experience it can be helpful to establish a perimeter around your planting area and patrol that regularly to prevent the ground ivy from crossing it.
You also may find the following control information helpful.