This paper proposes a detailed study of the Multi-Fragment (MF) tour construction (TC) algorithm for the Travelling Salesman Problem (TSP). This TC heuristic is based on an edge selection strategy which favors edges with the smallest cost under the constraint to have a feasible tour. Extensive computational experiments have been performed on benchmark instances from the literature. The results show that the studied algorithm generally outperforms other constructive heuristics for the TSP.