Simple, strict, proper, happy: A study of reachability in temporal graphs

Dynamic networks are a complex topic. Not only do they inherit the complexity of static networks (as a particular case) while making obsolete many techniques for these networks; they also happen to be deeply sensitive to specific definitional subtleties, such as strictness (can several consecutive edges be used at the same time instant?), properness (can adjacent edges be present at the same time?) and simpleness (can an edge be present more than once?). These features, it turns out, have a significant impact on the answers to various questions, which is a frequent source of confusion and incomparability among results. In this paper, we explore the impact of these notions, and of their interactions, in a systematic way. Our conclusions show that these aspects really matter. In particular, most of the combinations of the above properties lead to distinct levels of expressivity of a temporal graph in terms of reachability. Then, we advocate the study of an extremely simple model -- happy graphs -- where all these distinctions vanish. Happy graphs suffer from a loss of expressivity; yet, we show that they remain expressive enough to capture (and strengthen) interesting features of general temporal graphs. A number of questions are proposed to motivate the study of these objects further.