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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
##### Keywords:
Shannon capacity; independent domination number
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##### References:
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