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 $$D_k(G)$$ 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 $$D_{\varGamma (G)+1}(G)$$ D Γ ( G ) + 1 ( G ) is not necessarily connected, for $$\varGamma (G)$$ Γ ( 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 $$b \ge 3$$ b ≥ 3 . Moreover, we construct an infinite family of graphs such that $$D_{\gamma (G)+1}(G)$$ D γ ( G ) + 1 ( G ) has exponential diameter, for $$\gamma (G)$$ γ ( G ) the minimum size of a dominating set. On the positive side, we show that $$D_{n-\mu }(G)$$ D n - μ ( G ) is connected and of linear diameter for any graph G on n vertices with a matching of size at least $$\mu +1$$ μ + 1 .