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Some reconstruction problems arising in combinatorics and coding theory are motivated by applications in information transmission when the redundancy of messages is not sufficient for their exact reconstruction, and in molecular biology, when one is interested in reconstructing unknown genetic data, or in restoring an evolution process.
The commonly used representations of genetic data, such as genomes, are permutations and signed permutations. In this paper, we focus our attention on a survey of recent results concerning the reconstruction of permutations and signed permutations from their erroneous patterns which are distorted by transpositions or reversals that are global rearrangements of genomes and can be considered as biological errors on genomes. The proposed approach is based on the investigation of structural properties of corresponding Cayley graphs Γ(G,S) where the symmetric group Sn of permutations and the hyperoctahedral group of signed permutations are considered as a group G, and generating sets S are specified by two operations that are transpositions and reversals.
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