  The forcing number of a graph with a perfect matching $M$ is the minimum
number of edges in $M$ whose endpoints need to be deleted, such that the
remaining graph only has a single perfect matching. This number is of great
interest in theoretical chemistry, since it conveys information about the
structural properties of several interesting molecules. On the other hand, in
bipartite graphs the forcing number corresponds to the famous feedback vertex
set problem in digraphs.
  Determining the complexity of finding the smallest forcing number of a given
planar graph is still a widely open and important question in this area,
originally proposed by Afshani, Hatami, and Mahmoodian in 2004. We take a first
step towards the resolution of this question by providing an algorithm that
determines the set of all possible forcing numbers of an outerplanar graph in
polynomial time. This is the first polynomial-time algorithm concerning this
problem for a class of graphs of comparable or greater generality.
