Conversational recommender systems focus on the task of suggesting products to users based on the conversation flow. Recently, the use of external knowledge in the form of knowledge graphs has shown to improve the performance in recommendation and dialogue systems. Information from knowledge graphs aids in enriching those systems by providing additional information such as closely related products and textual descriptions of the items. However, knowledge graphs are incomplete since they do not contain all factual information present on the web. Furthermore, when working on a specific domain, knowledge graphs in its entirety contribute towards extraneous information and noise. In this work, we study several subgraph construction methods and compare their performance across the recommendation task. We incorporate pre-trained embeddings from the subgraphs along with positional embeddings in our models. Extensive experiments show that our method has a relative improvement of at least 5.62% compared to the state-of-the-art on multiple metrics on the recommendation task.