  Let $G$ be a graph and $F:V(G)\to2^N$ be a set function. The graph $G$ is
said to be \emph{F-avoiding} if there exists an orientation $O$ of $G$ such
that $d^+_O(v)\notin F(v)$ for every $v\in V(G)$, where $d^+_O(v)$ 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 $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ 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 $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ and $F(v)$
contains no two consecutive integers, then $G$ is $F$-avoiding. This partly
answers a problem proposed by Akbari et. al.(2020)

Combinatorics