  CSMA/CA networks have often been analyzed using a stylized model that is
fully characterized by a vector of back-off rates and a conflict graph.
Further, for any achievable throughput vector $\vec \theta$ the existence of a
unique vector $\vec \nu(\vec \theta)$ of back-off rates that achieves this
throughput vector was proven. Although this unique vector can in principle be
computed iteratively, the required time complexity grows exponentially in the
network size, making this only feasible for small networks.
  In this paper, we present an explicit formula for the unique vector of
back-off rates $\vec \nu(\vec \theta)$ needed to achieve any achievable
throughput vector $\vec \theta$ provided that the network has a chordal
conflict graph. This class of networks contains a number of special cases of
interest such as (inhomogeneous) line networks and networks with an acyclic
conflict graph. Moreover, these back-off rates are such that the back-off rate
of a node only depends on its own target throughput and the target throughput
of its neighbors and can be determined in a distributed manner.
  We further indicate that back-off rates of this form cannot be obtained in
general for networks with non-chordal conflict graphs. For general conflict
graphs we nevertheless show how to adapt the back-off rates when a node is
added to the network when its interfering nodes form a clique in the conflict
graph. Finally, we introduce a distributed chordal approximation algorithm for
general conflict graphs which is shown (using numerical examples) to be more
accurate than the Bethe approximation.
