Structure of cycles in Minimal Strong Digraphs

TL;DR

This paper analyzes the cycle structures in Minimal Strong Digraphs, revealing properties about strongly connected components, linear vertices, and their relationships within the graph's hierarchy.

Contribution

It introduces new theorems characterizing the structure of cycles and components in Minimal Strong Digraphs, including bounds on linear vertices and cycle length relationships.

Findings

Non trivial strongly connected components contain at least one linear vertex.

Each non trivial maximal or minimal vertex in the Hasse diagram has at least one linear vertex.

The number of linear vertices in a cycle is at least half its length, rounded down.

Abstract

This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after suppressing the arcs of the cycle. By this process and by the contraction of all SCs into single vertices we obtain a Hasse diagram from the MSD. Among other properties, we show that any SC conformed by more than one vertex (non trivial SC) has at least one linear vertex (a vertex with indegree and outdegree equal to 1) in the MSD (Theorem 1); that in the Hasse diagram at least one linear vertex exists for each non trivial maximal (resp. minimal) vertex (Theorem 2); that if an SC contains a number $\lambda$ of vertices of the cycle then it contains at least $\lambda$ linear vertices in the MSD (Theorem 3); and, finally, that given a cycle of length $q$ contained in the MSD, the number $\alpha$ of linear vertices contained in the MSD satisfies $\alpha \geq \lfloor (q+1)/2 \rfloor$ (Theorem 4).