Addition of new links in a communication network or formation of new social ties in a social network is an important way to improve the performance of the respective networks. In this letter, we consider the optimal link addition for a string network in order to minimize the average path length for the network. In prior work, the optimal addition of links to a string network has been found to lead to a network where all the added links are incident to a single anchor node. Furthermore, the position of the anchor node has been found to be fixed at either approximately 0.2 or 0.8 of the total number of nodes in the string network. In this letter, we provide an analytical justification for this observation and, in the process, identify the fixed fractional positions of the anchor nodes. We also discuss the significance of the anchor nodes for a string network, which is an important network model for several real-world communication and social networks.