A New Proof

We will show that for the case of h=0 and d=b-c, there is an O(n²) algorithm for sorting signed permutations by reversals. Our key idea is to prove (constructively) that the following condition is fulfilled for every step of the algorithm:

Condition 11.4.1   $\;$
There exists a reversal r, such that $b(\pi r)-c(\pi r)=b(\pi)-c(\pi)-1$, and the overlap graph of $\pi r$ does not contain unoriented components.

A vertex in the overlap graph, i.e., a gray edge e in the breakpoint graph, defines the reversal acting on the two black edges adjacent to e. Thus the effect of a reversal r on the overlap graph is as follows:

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Delete the vertex v that corresponds to the edge defining r.

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Complement the subgraph induced by v's neighbors, switching oriented edges by unoriented ones, and vice versa.

The choice of reversals needs to be a good one, e.g., one that maintains condition 11.4.1. We must therefore make sure no unoriented components are generated when applying the reversals. Such reversals are called safe.