A graph G on n vertices is arbitrarily partitionable (AP for short) if for every partition (λ1,...,λp) of n (that is, λ1+...+λp=n), the vertex set V(G) can be partitioned into p parts V1,...,Vp such that G[Vi] has order λi and is connected for every i∈{1,...,p}. Several aspects of AP graphs have been investigated to date, including structural and algorithmic aspects, and variants. This work is dedicated to minimal AP graphs, which are those AP graphs that are not spanned by any proper AP subgraph. In particular, we pursue previous investigations by Ravaux and Baudon, Przybyło, and Woźniak, who established that minimal AP graphs are not all trees, but conjectured that they should all be somewhat sparse. In that line, we investigate several aspects of minimal AP graphs, including their minimum degree, their maximum degree, and their clique number. Some of the results we establish arise from an exhaustive list we give of all minimal AP graphs of order at most 10. We also address new questions on the structure of minimal AP graphs.