Shifting paths to avoidable ones

An extension of an induced path P in a graph G is ′ such that deleting the endpoints results P. said to be avoidable if each its extensions contained cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured every contains k-vertex also same length, proved result for k = 2. The case 1 was known much earlier, due work Ohtsuki, Cheung, Fujisawa 1976. conjecture all 2020 by Bonamy, Defrain, Hatzel, Thiebaut. present paper, using similar approach, we strengthen their from reconfiguration point view. Namely, show graph, can transformed one sequence shifts, where two paths are shifts other union with + vertices. We obtain analogous not necessarily walks. contrast, statement cannot extended trails or isometric paths.