Reconfiguring Dominating Sets in Minor-Closed Graph Classes

Abstract For a graph G , two dominating sets D and $$D'$$ D ′ in non-negative integer k the set is said to -transform if there sequence $$D_0,\ldots ,D_\ell $$ 0 , … ℓ of such that $$D=D_0$$ = $$D'=D_\ell $$|D_i|\le k$$ | i ≤ k for every $$i\in \{ 0,1,\ldots ,\ell \}$$ ∈ { 1 } $$D_i$$ arises from $$D_{i-1}$$ - by adding or removing one vertex 1,\ldots . We prove some positive constant c are toroidal graphs arbitrarily large order n minimum -transforms only $$k\ge \max |D|,|D'|\}+c\sqrt{n}$$ ≥ max + c n Conversely, hereditary class $$\mathcal{G}$$ G has balanced separators $$n\mapsto n^\alpha ↦ α $$\alpha <1$$ < we C that, then $$k=\max |D|,|D'|\}+\lfloor Cn^\alpha \rfloor ⌊ C ⌋