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Let $G$ be a graph and $f: G\rightarrow G$ be a continuous map. We establisha structure theorem which describes the structures of the set$R(f)-\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set andthe periodic point set of $f$ respectively. Roughly speaking, the set$R(f)-\overline{P(f)}$ is covered by finitely many pairwise disjoint$f$-invariant open sets $U_{1\,},\,\cdots,\,U_{n\,}$; each $U_i$ contains aunique minimal set $W_i$ which absorbs each point of $U_i$; each $W_i$ lies infinitely many pairwise disjoint circles each of which is contained in aconnected closed set; all of these connected closed sets are contained in $U_i$and permutated cyclically by $f$. As applications of the structure theorem,several known results are improved or reproved.

Structures of $R(f)-\overline{P(f)}$ for graph maps $f$