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An analogue of the Shannon capacity of a graph. (English) Zbl 0717.05070
The Shannon capacity of a graph G is the value \(\alpha_ s(G)=\sup_{n}^ n\sqrt{\alpha (G^ n)}\), where \(\alpha (G^ n)\) is the independence number of the strong product of n copies of G. The independent domination number of G, denoted \({\mathcal K}(G)\), is the smallest possible cardinality of a set which is both independent and dominating. The author introduces an analogue of the Shannon capacity, namely the \({\mathcal K}\)-capacity \({\mathcal K}_ s(G)=\inf_{n}^ n\sqrt{{\mathcal K}(G^ n)}\). Consider the graph \(G_{{\mathcal A}}=(V,E)\) which has one vertex for each letter in an alphabet \({\mathcal A}\) and in which two vertices are adjacent if and only if the corresponding letters can be confused. The Shannon capacity of \(G_{{\mathcal A}}\) yields an upper bound on the cardinality of a set of n-letter words which are pairwise nonconfusable, and the \({\mathcal K}\)-capacity of \(G_{{\mathcal A}}\) yields a lower bound on the size of a maximal such set. The author makes use of linear programming duality to present lower bounds on the \({\mathcal K}\)- capacity and uses them to evaluate the \({\mathcal K}\)-capacity of certain cycles and trees.

MSC:
05C99 Graph theory
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