The structures of pointwise recurrent quasi-graph maps

We show that a continuous map $f$ from a quasi-graph $G$ to itself is pointwise recurrent if and only if one of the following two statements holds: (1) $X$ is a simple closed curve and $f$ is topologically conjugate to an irrational rotation on the unit circle ${\mathbb{S}}^1$; (2) $f$ is a perodic homeomorphism.