# Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair

## TL;DR

This paper constructs highly connected infinite directed graphs that lack edge-disjoint paths in both directions between a specific vertex pair, using self-similar graph techniques that may have broader applications.

## Contribution

It introduces a method to build highly connected digraphs with no edge-disjoint s-t and t-s paths, employing self-similar graph constructions.

## Key findings

- Constructed for all natural numbers k a k-edge-connected digraph
- Demonstrated the absence of edge-disjoint s-t and t-s paths in these graphs
- Utilized self-similar graphs as a key technique

## Abstract

We construct for all $ k\in \mathbb{N} $ a $ k $-edge-connected digraph $ D $ with $ s,t\in V(D) $ such that there are no edge-disjoint $ s \rightarrow t $ and $t\rightarrow s $ paths. We use in our construction "self-similar" graphs which technique could be useful in other problems as well.