Given a graph G and an integer k, a token addition and removal ({\sf TAR}
for short) reconfiguration sequence between two dominating sets Ds and
Dt of size at most k is a sequence S=⟨D0=Ds,D1…,Dℓ=Dt⟩ 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=Γ(G)+α(G)−1 (where Γ(G) is the maximum size of a minimal
dominating set and α(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 Kℓ-minor free graph as long as k≥Γ(G)+O(ℓlogℓ) and for planar graphs whenever k≥Γ(G)+3. Finally,
we show that if k=Γ(G)+tw(G)+1, then there also exists a linear
transformation between any pair of dominating sets.Comment: 13 pages, 6 figure