An index code for a broadcast channel with receiver side information is locally decodable if every receiver can decode its demand using only a subset of the codeword symbols transmitted by the sender instead of observing the entire codeword. Local decodability in index coding improves the error performance when used in wireless broadcast channels, reduces the receiver complexity and improves privacy in index coding. The locality of an index code is the ratio of the number of codeword symbols used by each receiver to the number message symbols demanded by the receiver. Prior work on locality in index coding have considered only single unicast and single-uniprior problems, and the optimal trade-off between broadcast rate and locality is known only for a few cases. In this paper we identify the optimal broadcast rate (including among non-linear codes) for a class of unicast problems with three or fewer receivers when the locality is equal to the minimum possible value, i.e., equal to one. The index code that achieves this optimal rate is based on a clique covering technique which is well known. The main contribution of this paper is in providing tight converse results by relating locality to broadcast rate, and showing that this known index coding scheme is optimal when locality is equal to one. Towards this we derive several structural properties of the side information graphs of three receiver unicast problems, and combine them with information-theoretic arguments to arrive at a converse. © 2020 IEEE.