[Paper Review] Some results on structure of all arc-locally (out) in-semicomplete digraphs

Lucas Ismaily Bezerra Freitas, Orlando Lee|arXiv (Cornell University)|Apr 22, 2021

Rings, Modules, and Algebras9 references4 citations

TL;DR

This paper characterizes the structure of connected arc-locally (out) in-semicomplete and arc-locally semicomplete digraphs, proving that such digraphs are either diperfect or odd extended cycles of length at least five. The results generalize prior work on strong arc-locally semicomplete digraphs and establish structural dichotomies based on forbidden subdigraphs and connectivity.

ABSTRACT

Arc-locally semicomplete and arc-locally in-semicomplete digraphs were introduced by Bang-Jensen as a common generalization of both semicomplete and semicomplete bipartite digraphs in 1993. Later, Bang-Jensen (2004), Galeana-Sanchez and Goldfeder (2009) and Wang and Wang (2009) provided a characterization of strong arc-locally semicomplete digraphs. In 2009, Wang and Wang characterized strong arc-locally in-semicomplete digraphs. In 2012, Galeana-Sanchez and Goldfeder provided a characterization of all arc-locally semicomplete digraphs which generalizes some results by Bang-Jensen. In this paper, we characterize the structure of arbitrary connected arc-locally (out) in-semicomplete digraphs and arbitrary connected arc-locally semicomplete digraphs.

Motivation & Objective

• To extend the characterization of arc-locally semicomplete digraphs beyond the strong case to arbitrary connected digraphs.
• To resolve the structural classification of connected arc-locally (out) in-semicomplete digraphs, which generalize tournaments and semicomplete bipartite digraphs.
• To unify and generalize previous results on arc-locally semicomplete digraphs by identifying structural dichotomies.
• To establish that such digraphs are either diperfect or induced by odd extended cycles of length at least five, providing a complete structural classification.

Proposed method

• Use of forbidden subdigraph characterization via orientations of $P_4$, specifically $H_1$, $H_2$, $H_3$, and $H_4$, to define arc-locally in-semicomplete and semicomplete classes.
• Application of strong component analysis and vertex partitioning to identify structural components such as clique cuts and special vertex sets.
• Employment of the concept of $\mathcal{P}$-dominance and $\mathcal{P}$-composition to model digraph extensions and hierarchical structures.
• Leverage directional duality: the inverse of an arc-locally semicomplete digraph is also arc-locally semicomplete, enabling symmetric reasoning.
• Use of extension operations and subdigraph containment to build and classify complex structures from basic components like $E_m$, $\overrightarrow{C_2}$, $\mathcal{C}^*_3$, and $TT_3$.
• Proof by contradiction and structural induction: if a vertex outside a strong component $Q$ (an odd extended cycle of length ≥5) dominates $Q$, then adjacency contradictions arise due to arc-local semicompleteness.

Experimental results

Research questions

• RQ1 What is the complete structural characterization of connected arc-locally in-semicomplete digraphs?
• RQ2 What is the complete structural characterization of connected arc-locally out-semicomplete digraphs?
• RQ3 What is the complete structural characterization of connected arc-locally semicomplete digraphs beyond the strong case?
• RQ4 Under what conditions does a connected arc-locally semicomplete digraph fail to be diperfect?
• RQ5 Can odd extended cycles of length at least five be the only non-diperfect arc-locally semicomplete digraphs?

Key findings

• A connected arc-locally (out) in-semicomplete digraph is either diperfect, admits a special vertex partition, or has a clique cut.
• A connected arc-locally semicomplete digraph is either diperfect or is an odd extended cycle of length at least five.
• If a connected arc-locally semicomplete digraph contains a strong component that induces an odd extended cycle of length at least five, then the entire vertex set is contained in that component.
• The class of arc-locally semicomplete digraphs includes all semicomplete and semicomplete bipartite digraphs, and the characterization extends previous results on strong instances.
• The inverse of an arc-locally semicomplete digraph is also arc-locally semicomplete, enabling directional duality in structural analysis.
• The characterization is tight: no other structural forms exist beyond diperfect digraphs and odd extended cycles of length at least five.