  For index coding problems with special structure on the side-information
graphs called Interlinked Cycle (IC) structures index codes have been proposed
in the literature (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for
Index Coding: Generalizing Cycles and Cliques", in \textit{IEEE Trans. Inf.
Theory, vol. 63, no. 6, Jun. 2017} with a correction in "Interlinked Cycles for
Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2
[cs.IT] 25 Feb 2018)). In this paper we consider a generalization of IC
structures called {\it IC structures with interlocked outer cycles}. For IC
structures with interlocked outer cycles we show that the optimal length (also
known as the minrank of the index coding problem) depends on the maximum number
of disjoint outer cycles. We give two sufficient conditions such that if any of
these is satisfied then we provide explicit optimal index code construction.
The conditions mentioned above are shown to be not necessary by an explicit
example.
