  A non-separating multicurve of a surface S of genus g with m punctures is a
multicurve c so that S-c is connected. For k>0 define the graph of
non-separting k-multicurves to be the graph whose vertices are non-separating
multicurves with k components and where two such multicurves are connected by
an edge if they can be realized disjointly. We show that if k is smaller than
g/2+1 then this graph is hyperbolic.
