Isometric copies of directed trees in orientations of graphs

TL;DR

This paper constructs finite graphs where every orientation contains all possible isometric copies of any directed tree with n vertices, and also shows that some orientations lack infinite diameter directed paths.

Contribution

It introduces a method to construct graphs with universal isometric copies of all directed trees of a given size and analyzes the existence of infinite diameter paths in orientations.

Findings

Constructed graphs contain all isometric directed trees of size n in any orientation.

Proved existence of orientations with no infinite diameter directed paths.

Evaluated the minimal size of such graphs.

Abstract

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $\omega$-paths of infinite diameter.