TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs

We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets Ds and Dt of G, in determining if there exists a sequence S=<D_1:=D_s,...,D_l:=D_t> of dominating sets of G such that for any two consecutive dominating sets Dr and D{r+1} with r<t, D{r+1}=(Dr\ u) U v, where uv is an edge of G. In a paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.