We are interested in the relation between the pathwidth of
a biconnected outerplanar graph and the pathwidth of its (geometric)
dual.
Bodlaender and Fomin, after having proved that the
pathwidth of every biconnected outerplanar graph is always at most
twice the pathwidth of its (geometric) dual plus two, conjectured
that there exists a constant c such that the pathwidth of every
biconnected outerplanar graph is at most c plus the pathwidth of
its dual. They also conjectured that this was actually true with c
being 1 for every biconnected planar graph. Fomin
proved that the second conjecture is true for all planar
triangulations. First, we construct for each p>0 a
biconnected outerplanar graph of pathwidth 2p+1 whose (geometric)
dual has pathwidth p+1, thereby disproving both conjectures.
Then we prove, in an algorithmic way, that the pathwidth of every
biconnected outerplanar graph is at most twice the pathwidth of its
(geometric) dual minus 1.
A tight interval for the studied relation is therefore obtained,
and we show that all the gaps within the interval actually happen.