On a lower bound for sorting signed permutations by reversals

Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,\ldots -n,n,n-1,\ldots 1)$ and let $$\tilde{s}=(0,a_1,a_2,\ldots a_n,-a_n,-a_{n-1},\ldots -a_1)$$ be any long cycle on the set $\{-n,-n+1,\ldots 0,1,\ldots n\}$. Then, $n$ and $a_n$ are always in the same cycle of the product $p\tilde{s}$. Furthermore, we show the new lower bound via plane permutations can be interpreted as the topological genera of orientable surfaces associated to signed permutations.