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Linear transformations between dominating sets in the TAR-model
arXiv - CS - Discrete Mathematics Pub Date : 2020-06-30 , DOI: arxiv-2006.16726 Nicolas Bousquet, Alice Joffard, Paul Ouvrard
arXiv - CS - Discrete Mathematics Pub Date : 2020-06-30 , DOI: arxiv-2006.16726 Nicolas Bousquet, Alice Joffard, Paul Ouvrard
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。
更新日期:2020-07-01
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。