Linear transformations between dominating sets in the TAR-model

TL;DR

This paper studies how to efficiently transform one dominating set into another in a graph using token addition and removal sequences, providing new bounds for various graph classes and parameters.

Contribution

It improves existing bounds for the existence of linear TAR reconfiguration sequences between dominating sets across different graph classes.

Findings

Linear TAR sequences exist when k = Γ(G) + α(G) - 1.

Extended results for K_ell-minor free and planar graphs with specific k bounds.

Existence of linear transformations when k = Γ(G) + tw(G) + 1.

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.