On the Structure of Interlinked Cycle Structures with Interlocked Outer Cycles

TL;DR

This paper analyzes the structural properties of interlinked cycle structures with interlocked outer cycles in index coding problems, providing an algorithm to identify maximum disjoint outer cycles, which influences optimal code length.

Contribution

It introduces a simple algorithm to find the maximum number of disjoint outer cycles in IC structures with interlocked outer cycles, enhancing understanding of their structure.

Findings

Structural properties of IC structures with interlocked outer cycles identified.

Algorithm for finding maximum disjoint outer cycles provided.

Insights into how outer cycle disjointness affects index code optimal length.

Abstract

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 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 Interlinked 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.