Reconfiguration of dominating sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of (D_k(G)), the graph consisting of a node for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that (D_{varGamma (G)+1}(G)) is not necessarily connected, for (varGamma (G)) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for (b ge 3). Moreover, we construct an infinite family of graphs such that (D_{gamma (G)+1}(G)) has exponential diameter, for (gamma (G)) the minimum size of a dominating set. On the positive side, we show that (D_{n-mu }(G)) is connected and of linear diameter for any graph G on n vertices with a matching of size at least (mu +1).