Graphs and Combinatorics
Abstract. Given a graph G, the k-dominating graph of G, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of Dk(G). In this paper we give conditions that ensure Dk(G) is connected.
Domination, Reconfiguration problems, Chordal graphs, Bipartite graphs
Haas, Ruth and Seyffarth, Karen, "The k-Dominating Graph" (2013). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.