In this paper, a collection of double integrator agents with bounded inputs is considered. Communication is possible between any two agents only if the inter-agent distance is less than a fixed threshold. A special node, referred to as the "leader," generates an unknown reference trajectory to which all the other agents are required to converge in the shortest possible time. Assuming the initial communication graph to be connected, a directed spanning tree rooted at the leader is identified using a local algorithm. The dynamics of any two agents connected by an edge in the selected tree are modeled as a time-optimal pursuitevasion game, while maintaining the inter-agent communication link. Using the corresponding feedback saddle-point strategies, local min-max time control laws for each pair of agents are formulated. For the selected tree, the proposed collection of local min-max time control strategies is shown to be the communication preserving global min-max time strategy.