Reconfiguration of dominating sets

Abstract

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 $\mu +1$.