Vertex partitions of chordal graphs

A k-tree is a chordal graph with no (k + 2)-clique. An l-tree-partition of a graph G is a vertex partition of G into 'bags,' such that contracting each bag to a single vertex gives an e-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k >= l >= 0, every k-tree has an l-tree-partition in which each bag induces a connected left perpendicular k/(e + 1) right perpendicular-tree. An analogous result is proved for oriented k-trees. (C) 2006 Wiley Periodicals, Inc.