Recently well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. However, as we illustrate in the paper, the homeomorphic embedding relation as it is usually de ned su ers from several inadequacies which make it less suitable in a logic programming context. We present several increasingly re ned ways to remedy this problem by providing more sophisticated treatments of variables and present a new, extended homeomorphic embedding relation. Keywords : Termination, Well-quasi orders, Logic Programming, Program Transformation, Program Analysis, Partial Deduction.