Three Propositions of the Milner-Welsh Theorem

Combinatorial method is a very effective method to study Karp conjecture. The combinatorial method was first used to prove that many graph properties are elusive. An important combinatorial technique developed during this period was summarized by Milner and Welsh as the Milner-Welsh Theorem, which has an important application in determining the complexity of decision trees of graphic properties. In this paper, we give two important propositions about the Milner-Welsh theorem, and use these propositions to judge the elusive of the connectivity of graphs.