论文标题
重新配置次要图形类中的主导集
Reconfiguring dominating sets in minor-closed graph classes
论文作者
论文摘要
对于图$ g $，两个主导套装$ d $和$ g $ in $ g $的$ d $，以及一个非负整数$ k $，如果有一个序列$ d_0，\ ldots，\ ldots，d_ \ ell $ g $ g $ g $ g $ d_ $ d_0 $ d_ $ d_0 $ d = d_0 $ d = d_0 $ d = d_0 $ d = d_0 $ | d_i | \ leq k $每$ i \ in \ {0,1，\ ldots，\ ell \} $，$ d_i $从$ d_ {i-1} $中添加或删除一个$ i \ in \ in \ in \ in \ in \ {1，\ ldots，\ ldots，\ ldots，\ ldots，\ el el \ ell \ ell \ ell \}。我们证明存在一些正常数$ c $，并且有圆环图$ g $是任意大订单$ n $的$ g $，并且两个最低统治套装$ d $和$ d $ in $ g $中的$ d $ d $ k $ transforms to $ d'$ d'$仅当$ k \ k \ geq \ geq \ geq \ geq \ geq \ geq \ geq \ max \ d | d | d | d | d | d'| d'|相反，对于每个具有$ n <1 $的订单$ n \ mapsto n^α$平衡分离器的$ {\ cal g} $，对于某些$α<1 $，我们证明，如果$ g $是$ {\ cal g} $ {\ cal g} $ n $ n $ n $ n $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d的$ g $，则有一些正常的$ c $ $ k $ -transforms to $ d'$ for $ k = \ max \ {| d |，| d'| \}+\ lfloor cn^α\ rfloor $。
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 $D=D_0$, $D'=D_\ell$, $|D_i|\leq k$ 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 $k\geq \max\{ |D|,|D'|\}+c\sqrt{n}$. Conversely, for every hereditary class ${\cal G}$ that has balanced separators of order $n\mapsto n^α$ for some $α<1$, we prove that there is some positive constant $C$ such that, if $G$ is a graph in ${\cal G}$ of order $n$, and $D$ and $D'$ are two dominating sets in $G$, then $D$ $k$-transforms to $D'$ for $k=\max\{ |D|,|D'|\}+\lfloor Cn^α\rfloor$.
