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.

翻译：具有完美匹配$M$的图的逼迫数是指，为了使得剩余图仅存在唯一完美匹配，需要从$M$中删除其端点所关联的最少边数。该数在理论化学中具有重要意义，因为它能揭示多种有趣分子的结构特性。另一方面，在二分图中，逼迫数对应于有向图中著名的反馈顶点集问题。确定给定平面图最小逼迫数的计算复杂性，仍是该领域一个广泛开放且重要的问题，该问题最初由Afshani、Hatami和Mahmoodian于2004年提出。我们通过提出一个能在多项式时间内确定外平面图所有可能逼迫数集合的算法，朝解决该问题迈出了第一步。这是针对具有相当或更大一般性的图类，关于该问题的首个多项式时间算法。