Reconfiguration Graphs for Dominating Sets

The dominating graph of a graph G has as its vertices all dominating sets of G, with an edge between two dominating sets if one can be obtained from the other by adding or deleting a single vertex of G. This is an example of a reconfiguration graph. This paper gives a brief introduction to the study of reconfiguration of dominating sets, and to the dominating graph. We highlight some previous results and present some new work. In particular we give new results on the existence of Hamilton paths in the dominating graph.