We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (Discrete Appl Math 231:95–112, 2017) and Hatanaka et al. (IEICE Trans Fundam Electron Commun Comput Sci 98(6):1168–1178, 2015). As a third example, we apply the method to the following well-studied problem: given two k-colorings $$\alpha $$α and $$\beta $$β of a graph G, can $$\alpha $$α be modified into $$\beta $$β by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and $$k=4$$k=4. By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for $$(k-2)$$(k-2)-connected chordal graphs.

中文翻译：

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使用收缩解图解决重新配置问题

我们在一般情况下介绍了解决重配置问题的动态规划方法。我们的方法基于收缩解图，该解图是通过执行一系列适当的边收缩从解图中获得的，这些边收缩减小了图的大小，而不会丢失解决所考虑的重新配置问题所需的任何关键信息。我们的通用框架捕捉了 Bonsma（Discrete Appl Math 231:95–112, 2017）和 Hatanaka 等人的已知重构结果背后的方法。(IEICE Trans Fundam Electron Commun Comput Sci 98(6):1168–1178, 2015)。作为第三个例子，我们将该方法应用于以下经过充分研究的问题：给定图 G 的两个 k-着色 $$\alpha $$α 和 $$\beta $$β，可以 $$\alpha $$α通过一次重新着色 G 的一个顶点，将其修改为 $$\beta $$β，同时保持k-着色？即使对于二部平面图和 $$k=4$$k=4，这个问题也是已知的 PSPACE-hard。通过将我们的方法与对图结构的彻底利用相结合，我们获得了 $$(k-2)$$(k-2) 连接弦图的多项式时间算法。