On digraphs without onion star immersions

TL;DR

This paper studies digraphs that do not contain a specific onion star immersion, revealing duality principles and establishing conditions involving arc-disjoint paths that imply the presence of such immersions.

Contribution

It introduces new duality results for digraphs excluding onion star immersions and provides functions linking arc-disjoint path structures to the existence of these immersions.

Findings

Existence of a function f(t) relating arc-disjoint paths to onion star immersion

Existence of a function g(t,k) connecting path multiplicities to immersion presence

Duality principles for digraphs excluding onion star immersions

Abstract

The $t$-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 $x$.