On Index coding for Complementary Graphs with focus on Circular Perfect Graphs

TL;DR

This paper characterizes the index coding broadcast rates for graphs whose complements are circular perfect, expanding the classes of graphs with known exact broadcast rates and analyzing relationships between a graph and its complement.

Contribution

It provides the broadcast rate for graphs with circular perfect complements and establishes bounds relating the broadcast rates of a graph and its complement.

Findings

Exact broadcast rate for graphs with circular perfect complements.

Bounds on the sum and product of broadcast rates of a graph and its complement.

Existence of circular perfect but imperfect graphs satisfying bounds with equality.

Abstract

Circular perfect graphs are those undirected graphs such that the circular clique number is equal to the circular chromatic number for each induced subgraph. They form a strict superclass of the perfect graphs, whose index coding broadcast rates are well known. We present the broadcast rate of index coding for side-information graphs whose complements are circular perfect, along with an optimal achievable scheme. We thus enlarge the known classes of graphs for which the broadcast rate is exactly characterized. In an attempt to understand the broadcast rate of a graph given that of its complement, we obtain upper and lower bounds for the product and sum of the vector linear broadcast rates of a graph and its complement. We show that these bounds are satisfied with equality even for some perfect graphs. Curating prior results, we show that there are circular perfect but imperfect graphs which satisfy the lower bound on the product of the broadcast rate of the complementary graphs with equality.