Interplays between variations of arbitrarily partitionable graphs under minimality constraints

An arbitrarily partitionable (AP) graph is a graph that can be partitioned into arbitrarily many connected graphs with arbitrary orders. Since independent seminal works by Barth, Baudon, and Puech, and Hor\v{n}\'{a}k and Wo\'{z}niak, AP graphs have been receiving increasing attention in literature, dedicated to understanding several of their aspects, including structural aspects, algorithmic aspects, and their connections with Hamiltonian graphs. Other aspects of interest cover variants of AP graphs, such as AP graphs that can be partitioned in an online way (OLAP graphs), AP graphs that can be partitioned in a recursive way (RAP graphs), and AP graphs that are edge-minimal (minAP graphs).In the current work, we initiate the study of the latter notion of minimality for OLAP and RAP graphs. That is, we wonder about OLAP and RAP graphs that are not spanned by any strictly smaller OLAP or RAP graph, respectively, leading to the notions of minOLAP and minRAP graphs. We prove that such non-trivial graphs exist, and explore connections between minAPness, minOLAPness, and minRAPness. In particular, we prove that some fundamental connections between APness, OLAPness, and RAPness do not generalise to their minimal counterparts. We also investigate small minOLAP and minRAP graphs, as well as sufficient conditions guaranteeing an OLAP or RAP graph is not minimal, thereby generalising known results on AP graphs. This work also includes many open questions and problems for further work on the topic, that are disseminated throughout.