Complexity of Hamiltonian Cycle Reconfiguration

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 $C_0,C_1,\dots ,C_t$ such that $C_i$ can be obtained from $C_{i-1}$ by a switch for each i with $1\le i\le t$ , where a switch is the replacement of a pair of edges $uv$ and $wz$ on a Hamiltonian cycle with the edges $uw$ and $vz$ of G, given that $uw$ and $vz$ 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.