Homomorphically Full Oriented Graphs

TL;DR

This paper extends the concept of homomorphically full graphs to oriented graphs, providing characterizations, recognition algorithms, and complexity results, including NP-completeness and GI-hardness, for various related problems.

Contribution

It introduces two new definitions of homomorphically full oriented graphs and analyzes their recognition complexity, including efficient algorithms and hardness results.

Findings

Homomorphically full oriented graphs are characterized as quasi-transitive orientations of homomorphically full graphs.

Recognition of these graphs can be efficient under certain definitions.

Deciding if a graph admits a homomorphically full orientation is NP-complete, and isomorphism of oriented cliques is GI-complete.

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.