  We study the PSPACE-complete $k$-Canadian Traveller Problem, where a weighted
graph $G=(V,E,\omega)$ with a source $s\in V$ and a target $t\in V$ are given.
This problem also has a hidden input $E_* \subsetneq E$ of cardinality at most
$k$ representing blocked edges. The objective is to travel from $s$ to $t$ with
the minimum distance. At the beginning of the walk, the blockages $E_*$ are
unknown: the traveller discovers that an edge is blocked when visiting one of
its endpoints. Online algorithms, also called strategies, have been proposed
for this problem and assessed with the competitive ratio, i.e. the ratio
between the distance actually traversed by the traveller divided by the
distance we would have traversed knowing the blockages in advance.
  Even though the optimal competitive ratio is $2k+1$ even on unit-weighted
planar graphs of treewidth 2, we design a polynomial-time strategy achieving
competitive ratio $9$ on unit-weighted outerplanar graphs. This value $9$ also
stands as a lower bound for this family of graphs as we prove that, for any
$\varepsilon > 0$, no strategy can achieve a competitive ratio $9-\varepsilon$.
Finally, we show that it is not possible to achieve a constant competitive
ratio (independent of $G$ and $k$) on weighted outerplanar graphs.
