To answer this complex question I would like you to visit the National Resources Conservation Service website that addresses the invasive plant issue in exquisite detail. For more specifics in your state, visit the Connecticut Invasive Plant Working Group website for lists of invasive plants and native alternatives for use in your landscape. What a great service they are providing for the citizens of Connecticut.
For eNature visitors from other states whose invasive plant information is not readily available, contact the powers that be and request it. One word of caution about research on invasive plants: An invasive plant in Maine may not be invasive in Ohio or Idaho, so make sure you make your own decisions based on local information. Check with your favorite native plant nursery or the Native Plant Society.