of subtrees An alternative characterization of chordal graphs, due to , involves trees and their subtrees. From a collection of subtrees of a tree, one can define a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed that the subtree graphs are exactly the chordal graphs. A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the junction tree algorithm. Relation to other graph classes Subclasses Interval graphs are the intersection graphs of subtrees of path graphs, a special case of trees. Therefore, they are a subfamily of chordal graphs. Split graphs are graphs that are both chordal and the complements of chordal graphs. showed that, in the limit as goes to infinity, the fraction of -vertex chordal graphs that are split approaches one. Ptolemaic graphs are graphs that are both chordal and distance hereditary. Quasi-threshold graphs are a subclass of Ptolemaic graphs that are both chordal and cographs. Block graphs are another subclass of Ptolemaic graphs in which every two maximal cliques have at most one vertex in common. A special type is windmill graphs, where the common vertex is the same for every pair of cliques. Strongly chordal graphs are graphs that are chordal and contain no -sun (for ) as an induced subgraph. Here an -sun is an -vertex chordal graph together with a collection of degree-two vertices, adjacent to the edges of a Hamiltonian cycle in . -trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size. Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees. Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal. Superclasses Chordal graphs are a subclass of the well known perfect graphs. Other superclasses of chordal graphs include weakly chordal graphs, cop-win graphs, odd-hole-free graphs, even-hole-free graphs, and Meyniel graphs. Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see