Arcwise increasing maps onto graphs

A continuous function $f$ from $[0,1]$ onto a Peano continuum $X$ is called arcwise increasing if for every two closed subintervals $A$ and $B$ of $[0,1]$ such that $A$ is a proper subset of $B$, then $f(A)$ is a proper subset of $f(B)$. A Peano continuum $X$ is said to admit an arcwise increasing map if there exists an arcwise increasing map $f:[0,1]\to X$. In this talk we characterize the finite graphs that admit an arcwise increasing map.