By Smelyakov S.V., Stoyan Y.G.

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Finally, shift again cyclically the first m coins (one time), reaching (c2, c1, c 3 , . . , cn-1, c1). Formally, it is easy to check that is allowed and 36 Chapter 2. Directed Graphs, Automata, and Automata Networks This shows that, for either m =n — 1 or m < n — 1, D penultimately realizes y with respect to 1 , . . , n. 5, we obtain again that D is penultimately permutation complete with respect ton. 6, the proof is complete. 10. Suppose that a strongly connected digraph D contains a strongly connected penultimately permutation complete subdigraph V having at least three vertices.

5 that (V \ {w}, S) is isomorphic to the degree (|V| — 1) complete transformation semigroup. , D is isomorphically (m — Incomplete and hence also isomorphically n-complete. Let D be a digraph. D. V is group n-complete (with respect to its semigroup) if the symmetric group on n letters divides the semigroup of D. 15. 16. Let D be a digraph containing all loop edges. Suppose that D has a strongly connected subdigraph with at least n + 1 vertices which contains a branch. Then D is isomorphically group n-complete.

Proof. The theorem is an immediate consequence of the Lagrange coordinate decomposition theorem and the Jordan-Holder theorem for finite groups. 1. Several books present various aspects of the basic notation and notions used in this book. Most are folklore going back to G. F. Frobenius at the turn of the nineteenth century. 2. 3 is from J. Denes and P. Hermann [1982]. Some other aspects of this result are given by Denes [1986]. 2, the other parts of this section should also be regarded as folklore.

### A basis of equivalence classes of paths in optimization problems by Smelyakov S.V., Stoyan Y.G.

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