Abstract

AbstractAn s, t Hamiltonian path P for an \(m \times n\) rectangular grid graph \(\mathbb {G}\) is a Hamiltonian path from the top-left corner s to the bottom-right corner t. Here we consider the reconfiguration of a subfamily of such s, t Hamiltonian paths called “simple” paths. We define an operation “square-switch” on simple paths P that affects only those edges of P that lie in some 2\(\,\times \,\)2 subgrid of \(\mathbb {G}\). We give an algorithmic proof that the Hamiltonian path graph \(\mathcal {G}\) for simple paths is connected for the square-switch operation: our algorithm reconfigures any given simple path P to any other given simple path \(P'\) in \(\mathcal {O}\)(|P|) time using at most 5|P|/4 square-switches, where \(|P| = m \times n\) is the number of vertices in a Hamiltonian path P for \(\mathbb {G}\). Thus the diameter of \(\mathcal {G}\) is at most 5mn/4 for square-switch, which we show is asymptotically tight.

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