https://hal-amu.archives-ouvertes.fr/hal-03882374Bożyk, ŁukaszŁukaszBożykMIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw] - UW - University of WarsawDefrain, OscarOscarDefrainMIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw] - UW - University of WarsawACRO - Algorithmique, Combinatoire et Recherche Opérationnelle - LIS - Laboratoire d'Informatique et Systèmes - AMU - Aix Marseille Université - UTLN - Université de Toulon - CNRS - Centre National de la Recherche ScientifiqueOkrasa, KarolinaKarolinaOkrasaMIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw] - UW - University of WarsawPilipczuk, MichałMichałPilipczukMIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw] - UW - University of WarsawOn digraphs without onion star immersionsHAL CCSD2022[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Defrain, Oscar2022-12-02 13:30:582023-03-24 14:53:292022-12-02 13:30:58enPreprints, Working Papers, ...1The $t$-onion star is the digraph obtained from a star with $2t$ leaves by replacing every edge by a triple of arcs, where in $t$ triples we orient two arcs away from the center, and in the remaining $t$ triples we orient two arcs towards the center. Note that the $t$-onion star contains, as an immersion, every digraph on $t$ vertices where each vertex has outdegree at most $2$ and indegree at most $1$, or vice versa. We investigate the structure in digraphs that exclude a fixed onion star as an immersion. The main discovery is that in such digraphs, for some duality statements true in the undirected setting we can prove their directed analogues. More specifically, we show the next two statements. There is a function $f\colon \mathbb{N}\to \mathbb{N}$ satisfying the following: If a digraph $D$ contains a set $X$ of $2t+1$ vertices such that for any $x,y\in X$ there are $f(t)$ arc-disjoint paths from $x$ to $y$, then $D$ contains the $t$-onion star as an immersion. There is a function $g\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ satisfying the following: If $x$ and $y$ is a pair of vertices in a digraph $D$ such that there are at least $g(t,k)$ arc-disjoint paths from $x$ to $y$ and there are at least $g(t,k)$ arc-disjoint paths from $y$ to $x$, then either $D$ contains the $t$-onion star as an immersion, or there is a family of $2k$ pairwise arc-disjoint paths with $k$ paths from $x$ to $y$ and $k$ paths from $y$ to $x$.