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 ℓ‐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 ${\lfloor k} /(\ell+1) \rfloor$ ‐tree. An analogous result is proved for oriented k ‐trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167–172, 2006