On the Structure of Interlinked Cycle Structures with Interlocked Outer Cycles.

For index coding problems with special structure on the side-information graphs called 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 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)). Recently (S. Sasi and B.S. Rajan, On Optimal Index Codes for Cycle Structures with Outer Cycles, in arxiv (arXiv:1804.09120v1 [cs.IT]), 24 Apr 2018) for a generalization of IC structures called IC structures with interlocked outer cycles optimal length index codes have been reported and it is shown that the optimal length depends on the maximum number of disjoint outer cycles. In this paper we discuss certain structural properties of IC structures with interlocked outer cycles and provide a simple algorithm to find the maximum number of disjoint outer cycles.