Linear transformations between dominating sets in the TAR-model
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Given a graph G and an integer k, a token addition and removal (TAR for short) reconfiguration sequence between two dominating sets D_ s and D_ t of size at most k is a sequence S= ⟨ D_0 = D_ s, D_1 …, D_ℓ = D_ t⟩ 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 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_ℓ-minor free graph as long as k ≥Γ(G)+O(ℓ√(logℓ)) 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.
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