On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has tree-partition-width at most $24k\Delta$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$ without the restriction that $k\geq3$

Bad News for Chordal Partitions

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question

Communication tree problems

In this paper, we consider random communication requirements and several cost measures for a particular model of tree routing on a complete network. First we show that a random tree does not give any approximation. Then give approximation algorithms for the case for two random models of requirements.Postprint (published version