Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page

Using contracted solution graphs for solving reconfiguration problems.

We introduce a dynamic programming method for solving reconfiguration problems, 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. As an example, we consider a 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? 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

Using Contracted Solution Graphs for Solving Reconfiguration Problems

We introduce a dynamic programming method for solving reconfiguration problems, 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. As an example, we consider a 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? 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

Using contracted solution graphs for solving reconfiguration problems

We introduce a dynamic programming method for solving reconfiguration problems, 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. As an example, we consider a 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? 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

A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.Comment: 22 pages, 5 figure

Reconfiguring Independent Sets in Claw-Free Graphs

We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an independent set. We also consider the more restricted model where $v$ and $w$ have to be adjacent