Generating strongly 2-connected digraphs

TL;DR

This paper proves that any strongly 2-connected digraph can be constructed from a small set of base digraphs using four specific operations, establishing a foundational method for generating such graphs.

Contribution

The paper introduces four operations that can generate all strongly 2-connected digraphs from a finite base set, advancing the understanding of their structural construction.

Findings

Existence of four operations for generating strongly 2-connected digraphs.

Any such digraph can be built from a finite base using these operations.

Provides a constructive method for understanding the structure of strongly 2-connected digraphs.

Abstract

We prove that there exist four operations such that given any two strongly $2$-connected digraphs $H$ and $D$ where $H$ is a butterfly-minor of $D$, there exists a sequence $D_0,\dots, D_n$ where $D_0=H$, $D_n=D$ and for every $0\leq i\leq n-1$, $D_i$ is a strongly $2$-connected butterfly-minor of $D_{i+1}$ which is obtained by a single application of one of the four operations. As a consequence of this theorem, we obtain that every strongly $2$-connected digraph can be generated from a concise family of strongly $2$-connected digraphs by using these four operations.