Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph

We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata and clarify their relation to endpoints of clique trees. Next we define a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as a bipartite graph and prove that the bipartite graph is always connected. Lastly we consider to characterize chordal graphs from the aspect of non-uniqueness of clique trees.