  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 all three receiver unicast problems 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
and 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.
