Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no dominating set has size bigger than $k$. We first improve a result of Haas and Seyffarth, by showing that if $k=\Gamma(G)+\alpha(G)-1$ (where $\Gamma(G)$ is the maximum size of a minimal dominating set and $\alpha(G)$ the maximum size of an independent set), then there exists a linear {\sf TAR} reconfiguration sequence between any pair of dominating sets. We then improve these results on several graph classes by showing that the same holds for $K_{\ell}$-minor free graph as long as $k \ge \Gamma(G)+O(\ell \sqrt{\log \ell})$ and for planar graphs whenever $k \ge \Gamma(G)+3$. Finally, we show that if $k=\Gamma(G)+tw(G)+1$, then there also exists a linear transformation between any pair of dominating sets.

中文翻译：

---

TAR 模型中支配集之间的线性变换

给定一个图 $G$ 和一个整数 $k$，两个支配集 $D_{\sf s}$ 和 $D_{\sf t}$ 之间的令牌添加和删除（简称 {\sf TAR}）重配置序列大小至多 $k$ 是 $G$ 的支配集的序列 $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ 使得任意两个连续支配集的区别在于增加或删除一个顶点，并且没有支配集的大小大于 $k$。我们首先改进了 Haas 和 Seyffarth 的结果，证明如果 $k=\Gamma(G)+\alpha(G)-1$（其中 $\Gamma(G)$ 是最小支配集的最大大小，并且$\alpha(G)$ 独立集的最大大小），那么在任何一对支配集之间都存在线性 {\sf TAR} 重配置序列。然后我们通过证明 $K_{\ell}$-minor free graph 只要 $k \ge \Gamma(G)+O(\ell \sqrt{\log \ ell})$ 和对于平面图，只要 $k \ge \Gamma(G)+3$。最后，我们证明如果 $k=\Gamma(G)+tw(G)+1$，那么在任何一对支配集之间也存在线性变换。