Linear transformations between dominating sets in the TAR-model

Abstract: 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.