Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homomorphically Full Oriented Graphs

Thomas Bellitto, Christopher Duffy, Gary MacGillivray

TL;DR

This work investigates homomorphically full properties for oriented graphs under two interpretations: antisymmetric digraphs and orientations of simple graphs. It provides a complete classification for antisymmetric digraphs: a digraph $G$ is homomorphically full iff it is quasi-transitive and $U(G)$ is homomorphically full, enabling polynomial-time recognition; for oriented graphs, fullness resists a simple criterion, with oriented cliques and the closure $cl(G)$ offering structural insights. The paper establishes GI-hardness for recognizing homomorphically full oriented graphs and NP-completeness for deciding whether a graph admits a homomorphically full orientation, via reductions involving oriented cliques and their isomorphism problems. It also shows that every orientation of a homomorphically full graph is full, while the converse does not hold, and outlines avenues for extending the theory to $2$-edge-coloured and reflexive cases as well as future research directions.

Abstract

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.

Homomorphically Full Oriented Graphs

TL;DR

This work investigates homomorphically full properties for oriented graphs under two interpretations: antisymmetric digraphs and orientations of simple graphs. It provides a complete classification for antisymmetric digraphs: a digraph is homomorphically full iff it is quasi-transitive and is homomorphically full, enabling polynomial-time recognition; for oriented graphs, fullness resists a simple criterion, with oriented cliques and the closure offering structural insights. The paper establishes GI-hardness for recognizing homomorphically full oriented graphs and NP-completeness for deciding whether a graph admits a homomorphically full orientation, via reductions involving oriented cliques and their isomorphism problems. It also shows that every orientation of a homomorphically full graph is full, while the converse does not hold, and outlines avenues for extending the theory to -edge-coloured and reflexive cases as well as future research directions.

Abstract

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.
Paper Structure (6 sections, 20 theorems, 2 equations, 3 figures)

This paper contains 6 sections, 20 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction and Background
  2. Homomorphically Full Antisymmetric Digraphs
  3. Homomorphically Full Oriented Graphs
  4. Deciding if $G$ is Homomorphically Full
  5. Orientations of Homomorphically Full Graphs
  6. Further Remarks and Future Work

Key Result

Theorem 1.1

B96 Let $\Gamma$ be a graph. The following statements are equivalent:

Figures (3)

  • Figure 1: A homomorphically full oriented graph with a pair of vertices that are not neighbourhood comparable
  • Figure 2: A homomorphically full oriented graph with a pair of vertices that are not neighbourhood comparable
  • Figure 3: An oriented clique on $7$ vertices.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • ...and 25 more