In other words, the node set of
is the set of gray edges in
,
and two nodes are connected by an arc if the intervals
associated with their gray edges overlap. We shall identify a
node in
with the edge it represents and with its
interval in the representation. Thus, the endpoints of a gray edge
are actually the endpoints of the interval representing the
corresponding node in
.
A connected component of
that contains an oriented edge is called an oriented component, otherwise, it is called an unoriented
component. Figure 10.10(c) shows the interval
overlap graph for
.
It has only one
oriented component. Figure 10.11(b) shows the
overlap graph of the permutation
,
which
has two connected components, one oriented and the
other unoriented.
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We shall see that any connected component which is oriented can be transformed by a series of reversals to a set of trivial connected components that correspond to the identity permutation. The unoriented connected components pose us a problem since we cannot split any of their cycles, nor delete any of their breakpoints by applying a single reversal.
In some cases we can eliminate unoriented components. This is done either by applying a reversal that does not increase the number of cycles, but rather transforms some of the edges to oriented edges, or by applying a reversal that merges two or more unoriented connected components into one oriented component.
The above idea for eliminating unoriented components allows a characterization of the unoriented components, on which we have to spend an extra reversal operation. We denote these components as hurdles.