Irregularity Notions for Digraphs

Locally irregular graphs are those graphs in which any two adjacent vertices have different degrees, while locally irregular decompositions are edge-partitions of graphs where each part induces a locally irregular graph. These notions were introduced in a seminal work of Baudon \textit{et al.}, in connection, in particular, to the so-called 1-2-3 Conjecture. Since then, several of their aspects of interest have been investigated in the literature, including the existence of such decompositions with few parts, and the complexity of finding such ones.In this work, we pursue investigations on generalisations of these notions to digraphs. In particular, we mainly investigate one variant in which the notion of irregularity for a digraph requires, for every arc from a vertex $u$ to a vertex $v$, the outdegree of $u$ be different from the indegree of $v$. We establish several results on this variant, covering upper bounds on the number of needed parts in decompositions, complexity aspects, and the impact of being able to choose the graph orientation, which results we compare to both the undirected setting and previous investigations on the directed one.