Using Contracted Solution Graphs for Solving Reconfiguration Problems

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 (2012) and Hatanaka, Ito and Zhou (2014). As a third example, we apply the method to the following problem: given two $k$-colorings $\alpha$ and $\beta$ of a graph $G$, can $\alpha$ be modified into $\beta$ 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 $k=4$. By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial time algorithm for $(k-2)$-connected chordal graphs.