On Index Codes for Interlinked Cycle Structured Side-Information Graphs

In connection with the index code construction and the decoding algorithm for interlinked cycle (IC) structures proposed by Thapa, Ong and Johnson in \cite{TOJ} ("Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017), it is shown in \cite{VaR} ("Optimal Index Codes For A New Class of Interlinked Cycle Structure", in \textit{IEEE Communication Letters,} available as early access article in \textit{IEEE Xplore}: DOI-10.1109/LCOMM.2018.2799202) that the decoding algorithm does not work for all IC structures. In this work, a set of necessary and sufficient conditions on the IC structures is presented for the decoding algorithm to work for the code construction given in \cite{TOJ}. These conditions are shown to be satisfied for the IC structures without any cycles consisting of only non-inner vertices.