Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Theoretical Computer Science - Game theory meets theoretical computer science
Randomly coloring sparse random graphs with fewer colors than the maximum degree
Random Structures & Algorithms
Relationships between nondeterministic and deterministic tape complexities
Journal of Computer and System Sciences
Mixing 3-colourings in bipartite graphs
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
The connectivity of boolean satisfiability: computational and structural dichotomies
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
On the complexity of reconfiguration problems
Theoretical Computer Science
Shortest paths between shortest paths and independent sets
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Approximability of the subset sum reconfiguration problem
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
An improved sufficient condition for reconfiguration of list edge-colorings in a tree
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Complexity of independent set reconfigurability problems
Theoretical Computer Science
Reconfiguration of list edge-colorings in a graph
Discrete Applied Mathematics
The complexity of rerouting shortest paths
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
The complexity of rerouting shortest paths
Theoretical Computer Science
Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs
Journal of Combinatorial Optimization
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Suppose we are given a graph G together with two proper vertex k-colourings of G, @a and @b. How easily can we decide whether it is possible to transform @a into @b by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k=2, and decidable in polynomial time for k=3. Here we prove it is PSPACE-complete for all k=4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4@?k@?6, and (ii) bipartite planar graphs and k=4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour @a to @b. We also exhibit, for every k=4, a class of graphs {G"N","k:N@?N^*}, together with two k-colourings for each G"N","k, such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is @W(2^N), whereas the size of G"N is O(N^2). This is in stark contrast to the k=3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4@?k@?6 planar graphs and (ii) for k=4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.