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If i=j, the second sequence is empty. Create a new pair of darts d, d’ and insert the pair as a new orbit into α. ,dj, d’) which are inserted into the set of orbits. For the inverse operation edge contraction, the steps are reversed. The edge to be contracted must not be a self-loop. By definition, the operator increases the number of orbits in the σ and α permutation by one. The number of components and ϕ orbits does not change since the newly inserted edge ensures that the graph of the map remains connected (proof in [8]).

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Leonardis, and H. Bischof. Hierarchical, adaptive and robust methods for image understanding. Surveys on Mathematics for Industry, No. 9:pp. 1–47, 1999. [29] W. G. Kropatsch and H. Macho. Finding the structure of connected components using dual irregular pyramids. In Cinqui`eme Colloque DGCI, pages 147–158. LLAIC1, Universit´e d’Auvergne, ISBN 2-87663-040-0, September 1995. [30] S. G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.

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