K-tree

The k-trees are exactly the maximal graphs with a treewidth of k ("maximal" means that no more edges can be added without increasing their treewidth).^[2] They are also exactly the chordal graphs all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k.^[1]

1-trees are the same as trees. 2-trees are maximal series–parallel graphs,^[3] and include also the maximal outerplanar graphs. Planar 3-trees are also known as Apollonian networks.^[4]

The graphs that have treewidth at most k are exactly the subgraphs of k-trees, and for this reason they are called partial k-trees.^[2]

The graphs formed by the edges and vertices of k-dimensional stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope, are k-trees when k ≥ 3.^[5] This gluing process mimics the construction of k-trees by adding vertices to a clique.^[6] A k-tree is the graph of a stacked polytope if and only if no three (k + 1)-vertex cliques have k vertices in common.^[7]