For a positive integer k, a vertex is said to be k-simplicial if its neighborhood is a k-clique. The smallest k-tree is a clique with k vertices. A graph G with more than k vertices is said to be k-tree if it has a k-simplicial vertex v such that G -- v is a k-tree. Note that the 1-trees are the class of trees. In graph minor theory, the k-trees are closely related to the concept of tree-width....