Reconfiguring Dominating Sets in Minor-Closed Graph Classes

Abstract

For a graph G, two dominating sets D and D' in G, and a non-negative integer k, the set D is said to k-transform to D' if there is a sequence D_0,ldots ,D_ell of dominating sets in G such that D=D_0, D'=D_ell , |D_i|le k for every iin { 0,1,ldots ,ell }, and D_i arises from D_{i-1} by adding or removing one vertex for every iin { 1,ldots ,ell }. We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and D' in G such that Dk-transforms to D' only if kge max { |D|,|D'|}+csqrt{n}. Conversely, for every hereditary class mathcal{G} that has balanced separators of order nmapsto n^alpha for some alpha <1, we prove that there is some positive constant C such that, if G is a graph in mathcal{G} of order n, and D and D' are two dominating sets in G, then Dk-transforms to D' for k=max { |D|,|D'|}+lfloor Cn^alpha rfloor .