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

对于一个图形G ^，二支配集d和\（d“\）在ģ，和一个非负整数ķ，设定d据说ķ -transform到\（d” \） ，如果有一个序列\（ D_0，\ ldots，D_ \ ELL \）在主导套ģ使得\（d = D_0 \） ，\（d“= D_ \ ELL \） ，\（| D_i | \文件ķ\）每\（ i \ in \ {0,1, \ ldots, \ell \} \)和\ (D_i \)由\ (D_ {i-1} \)通过为每个\ (i \ in \ {1, \ ldots, \ ell \} \). 我们证明了有一些正的常数Ç和有环形图ģ任意大的顺序的Ñ两个最小支配集，和d和\（d“\）在ģ使得d ķ -transforms到\（d” \）只如果\ (k \ ge \ max \ {| D |, | D '| \} + c \ sqrt {n} \)。相反，对于每个具有平衡顺序分隔符\ (n \ mapto n ^ \ alpha \)对于某些\ (\ alpha <1 \) 的遗传类\ (\ mathcal {G} \)，我们证明存在一些正常数C使得，如果G是一个图\ (\ mathcal {G} \)阶n，并且D和\ (D '\)是G中的两个支配集，然后D k 转换为\ (D' \)对于\ (k = \ max \ { | D |, | D'| \} + \ lfloor Cn ^ \ alpha \ rfloor \)。