Abstract
The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.
Highlights
A reconfiguration problem asks, given two feasible solutions of a combinatorial problem together with some transformation rules between the solutions, whether there is a step-by-step transformation from one solution to the other such that all intermediate states are feasible
Given two Hamiltonian cycles C0 and Ct of a graph G, the Hamiltonian cycle reconfiguration problem asks whether there is a sequence of Hamiltonian cycles C0, C1, . . . , Ct such that Ci and Ci+1 differ in two edges for each i with 0 ≤ i < t
Unit interval graphs form a proper subclass of strongly chordal graphs, and bipartite permutation graphs form a proper subclass of chordal bipartite graphs (See [13] for example)
Summary
A reconfiguration problem asks, given two feasible solutions of a combinatorial problem together with some transformation rules between the solutions, whether there is a step-by-step transformation from one solution to the other such that all intermediate states are feasible. Given two Hamiltonian cycles C0 and Ct of a graph G, the Hamiltonian cycle reconfiguration problem asks whether there is a sequence of Hamiltonian cycles C0 , C1 , . The Hamiltonian cycle reconfiguration problem can be defined in terms of the transformation rule, which is called switch (Switches are used for sampling and counting perfect matchings [4,5] and transforming graphs with the same degree sequence ([6,7], p.46)). The Hamiltonian cycle problem, which asks whether a given graph has a Hamiltonian cycle, is one of the well-known NP-complete problems [9], but the complexity of its reconfiguration version still seems to be open
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