Total well dominated trees
Arthur S. Finbow, Allan Frendrup, Preben Dahl Vestergaard · 2009
Let G = (V,E) be a graph with no isolated vertex. A set D is called total dominating in G if each vertex in G is adjacent to a vertex from D, and D is a minimal total dominating set if any subset D′ ⊂ D is not a total dominating set in G. If all minimal total dominating sets in G have the same cardinality then G is a total well dominated graph. In this paper we study composition and decomposition of total well dominated trees. By a reversible process we prove that any total well dominated tree can both be reduced to and constructed from a family of three small trees.