Vertex partitions of chordal graphs

A k‐tree is a chordal graph with no (k + 2)‐clique. An ?‐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 ?‐tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ? ≥ 0, every k‐tree has an ?‐tree‐partition in which each bag induces a connected ‐tree. An analogous result is proved for oriented k‐trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167–172, 2006