Smaller edge augmentations yielding Hamiltonicity or pancyclicity

Determine whether a smaller addition of token-sliding edges to the dominating graph can induce Hamiltonicity or pancyclicity comparable to that obtained by including the token-sliding edges used in the TARS-graph.

Background

The TARS-graph is formed by adding token-sliding adjacencies to the token addition/removal dominating graph. The paper shows that this combined rule set produces pancyclic graphs for many seed graphs, even though dominating graphs themselves never have Hamilton cycles.

The proofs use only a small number of token-sliding edges at any one time, motivating the unresolved question of whether fewer added edges could suffice to produce Hamiltonicity or pancyclicity.