The following occurred to me while playing Sokoban.

Let $G=(V,E)$ be a finite, simple, undirected, and connected graph with $|V|>1$. We call a function $f:V\to V$ a *push function* if $\{x,f(x)\}\in E$ for all $x\in V$. (Note that a push function cannot have fixed points, and that the composition of two push functions is usually not a push function.)

We say that $G$ is *contractible* if there is $n\in \mathbb{N}$ and there are push functions $f_1,\ldots,f_n:V\to V$ such that $$\text{im}(f_n\circ \ldots \circ f_1) = \{v\} \text{ for some } v\in V.$$

I think I have been able to prove that if a connected graph $G$ with more than 1 vertex is contractible then $G$ contains an odd cycle. Does the converse hold?