Vertex Partitions of Chordal Graphs

A \emph{$k$-tree} is a chordal graph with no $(k+2)$-clique. An \emph{$\ell$-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 $\ell$-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all $k\geq\ell\geq0$, every $k$-tree has an $\ell$-tree-partition in which every bag induces a connected $\floor{k/(\ell+1)}$-tree. An analogous result is proved for oriented $k$-trees.