MVP parking functions, permutation subgraphs and Motzkin paths

In parking problems, a given number of cars enter a one-way street sequentially,
and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars have the same preferred spot. We study a recent model introduced by Harris, Kamau, Mori, and Tian in recent work, called the MVP parking problem. In this model, priority is given to the cars arriving later in the sequence. When a car finds its preferred spot occupied by a previous car, it “bumps” that car out of the spot and parks there. The earlier car then has to drive on, and parks in the first available spot it can find. If all cars manage to park through this procedure, we say that the list of preferences is a (MVP) parking function.
We study the outcome map of MVP parking functions, which describes in what order the cars end up. In particular, we link the fibres of the outcome map to certain subgraphs of the inversion graph of the outcome permutation. This allows us to reinterpret and improve bounds from Harris et al. on the fibre sizes. We also focus on a subset of parking functions, called Motzkin parking functions, where every spot is preferred by at most two cars. We generalise results from Harris et al., and exhibit rich connections to Motzkin paths.