Minimum number of token-sliding edges required for pancyclicity

Determine how many token-sliding edges must be added to the dominating graph of a seed graph to obtain a pancyclic reconfiguration graph.

Background

The paper observes that the dominating graph is in some sense already close to being Hamiltonian and that the constructions in Sections 2 and 3 use only a small number of token-sliding edges at a time.

No general threshold or exact requirement is established for the number of token-sliding edges needed to make a dominating reconfiguration graph pancyclic. Determining this quantity would measure the augmentation required to obtain the stronger cycle-structure property.