Trees with domination subdivision number one.
Stephen Benecke, Christina M. Mynhardt · Australas. J Comb. · 2008
The domination subdivision number sdγ(G) of a graph G is the minimum number of edges that must be subdivided to increase the domination number of G. We present a simple characterization of trees with sdγ = 1 and a fast algorithm to determine whether a tree has this property.