The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $lambda_n=1/binom{n+1}{2}$. For $lambda_n=(1-epsilon)/binom{n+1}{2}$, the random graph consists of components of size at most $O(nln(n))$ a.s. and for $(1+epsilon)/binom{n+1}{2}$, there emerges a unique largest component of size $sim wp(epsilon) cdot 2^ncdot n$!$ a.s.. This giant component is furthermore dense in the reversal graph.
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