Using contracted solution graphs for solving reconfiguration problems

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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 \(\alpha \) 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.

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  1. 1.

    In [8], it is first shown that the list-coloring version of the \(\mathcal {C}_k\)-Reachability problem is PSPACE-complete for connected planar bipartite 4-colorable graphs. In the list-coloring version every vertex is assigned a list of two or three allowed colors. In the next step of the transformation, to the \(\mathcal {C}_k\)-Reachability problem, the list-coloring constraints are simulated by making every vertex adjacent to appropriately colored vertices of a small bipartite graph with a frozen 4-coloring. A frozen k-coloring is a k-coloring where every vertex is adjacent to vertices of each other color, so no vertex can be recolored. Since we do not care about planarity, we can instead simply take one highly connected bipartite graph with a frozen k-coloring and link all vertices of the first graph to the appropriate vertices of this new graph, to enforce the list-color constraints, while maintaining bipartiteness and ensuring \((k-2)\)-connectedness.

  2. 2.

    Since node names are irrelevant, we will simply write \((H,\ell )=\mathcal {S}^c(G,T)\) to denote that there is a label preserving isomorphism between the two. More formally, \(\mathcal {S}^c(G,T)\) can be seen as a class of labeled graphs that are equivalent under labeled isomorphisms.

  3. 3.

    It is possible for the given example to choose two solutions \(\alpha \) and \(\beta \), and correctly mark an \(\alpha \)-node x with respect to a one certificate \(S^1\), and a \(\beta \)-node y with respect to another certificate \(S^2\), such that \(\alpha \) and \(\beta \) are in different components of \(\mathcal {S}(G)\), but x and y are in the same component of \(\mathcal {S}^c(G,T)\). This is clearly not desirable; see Lemma 2.


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The authors would like to thank Carl Feghali and Matthew Johnson for fruitful discussions and two anonymous reviewers for helpful comments.

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Correspondence to Daniël Paulusma.

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An extended abstract of this paper appeared in the proceedings of MFCS 2016 [11].

Paul Bonsma: Supported by the European Community’s Seventh Framework Programme (FP7/2007-2013), Grant agreement No. 317662. Daniël Paulusma: Supported by EPSRC Grant EP/K025090/1.

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Bonsma, P., Paulusma, D. Using contracted solution graphs for solving reconfiguration problems. Acta Informatica 56, 619–648 (2019).

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