  In this paper we present some new complexity results on the routing time of a
graph under the \textit{routing via matching} model. This is a parallel routing
model which was introduced by Alon et al\cite{alon1994routing}. The model can
be viewed as a communication scheme on a distributed network. The nodes in the
network can communicate via matchings (a step), where a node exchanges data
(pebbles) with its matched partner. Let $G$ be a connected graph with vertices
labeled from $\{1,...,n\}$ and the destination vertices of the pebbles are
given by a permutation $\pi$. The problem is to find a minimum step routing
scheme for the input permutation $\pi$. This is denoted as the routing time
$rt(G,\pi)$ of $G$ given $\pi$. In this paper we characterize the complexity of
some known problems under the routing via matching model and discuss their
relationship to graph connectivity and clique number. We also introduce some
new problems in this domain, which may be of independent interest.
