  Kleinberg introduced three natural clustering properties, or axioms, and
showed they cannot be simultaneously satisfied by any clustering algorithm. We
present a new clustering property, Monotonic Consistency, which avoids the
well-known problematic behaviour of Kleinberg's Consistency axiom, and the
impossibility result. Namely, we describe a clustering algorithm, Morse
Clustering, inspired by Morse Theory in Differential Topology, which satisfies
Kleinberg's original axioms with Consistency replaced by Monotonic Consistency.
Morse clustering uncovers the underlying flow structure on a set or graph and
returns a partition into trees representing basins of attraction of critical
vertices. We also generalise Kleinberg's axiomatic approach to sparse graphs,
showing an impossibility result for Consistency, and a possibility result for
Monotonic Consistency and Morse clustering.
