Let G be a graph and F:V(G)β2N be a set function. The graph G is
said to be \emph{F-avoiding} if there exists an orientation O of G such
that dO+β(v)β/F(v) for every vβV(G), where dO+β(v) denotes the
out-degree of v in the directed graph G with respect to O. In this paper,
we give a Tutte-type good characterization to decide the F-avoiding problem
when for every vβV(G), β£F(v)β£β€21β(dGβ(v)+1) and F(v)
contains no two consecutive integers. Our proof also gives a simple polynomial
algorithm to find a desired orientation. As a corollary, we prove the following
result: if for every vβV(G), β£F(v)β£β€21β(dGβ(v)+1) and F(v)
contains no two consecutive integers, then G is F-avoiding. This partly
answers a problem proposed by Akbari et. al.(2020