Reconfiguring Dominating Sets in Minor-Closed Graph Classes

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