From expanded digraphs to lifts of voltage digraphs and line digraphs

TL;DR

This paper introduces a unified framework for constructing large digraphs from smaller ones, highlighting the relationships between lifted digraphs, line digraphs, and voltage digraphs, with applications to De Bruijn and Kautz graphs.

Contribution

It establishes the equivalence of various digraph construction techniques and provides conditions for when lifted digraphs retain line digraph properties, advancing graph theory methods.

Findings

Demonstrates the relationship between lifted digraphs and line digraphs.

Shows De Bruijn and Kautz graphs as lifts of smaller graphs.

Provides conditions for lifted digraphs to be line digraphs.

Abstract

In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line digraphs, which are two known ways to obtain dense digraphs. In the same context, we show the equivalence between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, the De Bruijn and Kautz digraphs, where it is shown that both families can be seen as lifts of smaller De Bruijn digraphs with appropriate voltage assignments.