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Algorithms 2018, 11(9), 140; https://doi.org/10.3390/a11090140

Complexity of Hamiltonian Cycle Reconfiguration

Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Rokkakubashi 3-27-1 Kanagawa-ku, Yokohama-shi, Kanagawa 221-8686, Japan
Received: 25 February 2018 / Revised: 17 August 2018 / Accepted: 14 September 2018 / Published: 17 September 2018
(This article belongs to the Special Issue Reconfiguration Problems)
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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. View Full-Text
Keywords: bipartite permutation graphs; chordal bipartite graphs; combinatorial reconfiguration; Hamiltonian cycle; PSPACE-complete; split graphs; strongly chordal graphs; unit interval graphs bipartite permutation graphs; chordal bipartite graphs; combinatorial reconfiguration; Hamiltonian cycle; PSPACE-complete; split graphs; strongly chordal graphs; unit interval graphs
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Takaoka, A. Complexity of Hamiltonian Cycle Reconfiguration. Algorithms 2018, 11, 140.

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