An Analogue of the Shannon Capacity of a Graph

Abstract

The Shannon capacity of a graph G is the value $\alpha _s ( G ) = \sup_n \sqrt[n]{\alpha ( G^n )}$, where $\alpha ( G^n )$ is the independence number of the strong product of n copies of G. We introduce an analogue of the Shannon capacity, namely $\kappa_s ( G ) = \inf_n \sqrt[n]{\kappa ( G^n )}$, where $\kappa ( G^n )$ is the independent domination number of the strong product of n copies of G. The Shannon capacity measures how rich a language can be, where the language is to be transmitted through a noisy channel. The parameter $\kappa _s $, on the other hand, measures how sparse such a language can be, if it is maximal with respect to inclusion.