The topological interference management (TIM) problem is a framework for studying partially connected interference networks with no channel state information at transmitters (CSIT), except network topology. TIM is a more pragmatic setting as CSIT is often available imperfectly, or may not be available at all. In this paper, we study the TIM problem with confidential messages, denoted in short by the secure TIM (STIM) problem. More specifically, we focus on the STIM problem for half-rate-feasible (HRF) networks. Half-rate-feasible networks are a class of partially connected interference networks whose sum degrees of freedom (DoF) have been characterized by K/2, without any secrecy constraints. The main contribution of this paper is as follows: We design achievable schemes for HRF networks subject to secrecy constraints, and present a lower bound on the secure degrees of freedom (SDoF). To this end, we first show the necessity of classifying HRF networks into two sub-categories based on some properties of the underlying network topology. As it turns out, the division of HRF networks into these sub-categories is critical for the design of secure transmission schemes. We then leverage the underlying topological properties along with ideas from secure interference alignment (SIA) in order to design achievable schemes for both subclasses of HRF networks.