eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
37:1
37:14
10.4230/LIPIcs.ISAAC.2020.37
article
Linear Transformations Between Dominating Sets in the TAR-Model
Bousquet, Nicolas
1
Joffard, Alice
1
Ouvrard, Paul
2
CNRS, LIRIS, Université de Lyon, Université Claude Bernard Lyon 1, France
Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, 33400 Talence, France
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 = ⟨ D₀ = D_s, D₁ …, 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 [R. Haas and K. Seyffarth, 2017], 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 √(log 𝓁)) 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.37/LIPIcs.ISAAC.2020.37.pdf
reconfiguration
dominating sets
addition removal
connectivity
diameter
minor
treewidth