Reconfiguration of Dominating Sets
Akira Suzuki, Amer E. Mouawad, Naomi Nishimura · arXiv (Cornell University) · 2014
We explore a reconguration 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 Dk(G), the graph con-sisting of a vertex for each dominating set of size at most k and edges speci ed by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D (G)+1(G) is not neces-sarily connected, for (G) the maximum cardinality of a minimal dom-inating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for b 3. Moreover, we construct an infinite family of graphs such that D (G)+1(G) has exponential diameter, for (G) the minimum size of a dominating set. On the positive side, we show that Dn(G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least + 1.