 
Summary: SIAM J. COMPUT. c 2007 Society for Industrial and Applied Mathematics
Vol. 36, No. 6, pp. 17481763
WHOLE GENOME DUPLICATIONS AND CONTRACTED
BREAKPOINT GRAPHS
MAX A. ALEKSEYEV AND PAVEL A. PEVZNER
Abstract. The genome halving problem, motivated by the whole genome duplication events
in molecular evolution, was solved by ElMabrouk and Sankoff in the pioneering paper [SIAM J.
Comput., 32 (2003), pp. 754792]. The ElMabroukSankoff algorithm is rather complex, inspiring
a quest for a simpler solution. An alternative approach to the genome halving problem based on
the notion of the contracted breakpoint graph was recently proposed in [M. A. Alekseyev and P.
A. Pevzner, IEEE/ACM Trans. Comput. Biol. Bioinformatics, 4 (2007), pp. 98107]. This new
technique reveals that while the ElMabroukSankoff result is correct in most cases, it does not
hold in the case of unichromosomal genomes. This raises a problem of correcting a flaw in the El
MabroukSankoff analysis and devising an algorithm that deals adequately with all genomes. In this
paper we efficiently classify all genomes into two classes and show that while the ElMabroukSankoff
theorem holds for the first class, it is incorrect for the second class. The crux of our analysis is a new
combinatorial invariant defined on duplicated permutations. Using this invariant we were able to
come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome
halving problem.
Key words. genome duplication, genome halving, genome rearrangement, breakpoint graph,
