  Given an undirected graph G = (V, E) with n vertices, and a function f : V ->
N, we consider the problem of finding a connected f -factor in G. In this work
we design an algorithm to check for the existence of a connected f -factor, for
the case where f (v) >= n/g(n), for all v in V and g(n) is polylogarithmic in
n. The running time of our algorithm is O(n^{2g(n)}. As a consequence of this
algorithm we conclude that the complexity of connected f -factor for the case
we consider is unlikely to be NP-Complete unless the Exponential Time
Hypothesis (ETH) is false. Secondly, under the assumption of the ETH, we show
that it is also unlikely to be in P for g(n) in O((log n)^{1+eps} ) for any
eps> 0. Therefore, our results show that for all eps> 0, connected f -factor
for f (v) >= n/O(log n)^{1+eps}) is in NP-Intermediate unless the ETH is false.
Further, for any constant c > 0, when g(n) = c, our algorithm for connected f
-factor runs in polynomial time. Finally, we extend our algorithm to compute a
minimum weight connected f -factor in edge weighted graphs in the same
asymptotic time bounds.
