Extending Ghouila-Houri's Characterization of Comparability Graphs to Temporal Graphs

2025-10-08

AI Summary

This paper generalizes Ghouila-Houri’s classical characterization of comparability graphs to temporal graphs—the first such extension—systematically addressing the modeling and recognition of comparability structures in temporal settings. We introduce the notion of *temporal transitive orientation*, establishing its equivalence to quasi-transitive orientations on underlying static graphs. For both single- and multi-label temporal graphs, we formulate equivalent 2-SAT constraints and design linear-time recognition algorithms. Furthermore, we derive a complete combinatorial characterization of temporal comparability graphs via forbidden temporally ordered subpatterns. Our main contributions are: (1) the first temporal analogue of the Ghouila-Houri theorem; (2) a structural characterization via 2-SAT and a linear-time decision algorithm; and (3) a proof that all results extend to multi-temporal-label settings, thereby broadening the applicability of static comparability theory to dynamic graph models.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

An orientation of a given static graph is called transitive if for any three vertices $a,b,c$, the presence of arcs $(a,b)$ and $(b,c)$ forces the presence of the arc $(a,c)$. If only the presence of an arc between $a$ and $c$ is required, but its orientation is unconstrained, the orientation is called quasi-transitive. A fundamental result presented by Ghouila-Houri guarantees that any static graph admitting a quasi-transitive orientation also admits a transitive orientation. In a seminal work, Mertzios et al. introduced the notion of temporal transitivity in order to model information flows in simple temporal networks. We revisit the model introduced by Mertzios et al. and propose an analogous to Ghouila-Houri's characterization for the temporal scenario. We present a structure theorem that will allow us to express by a 2-SAT formula all the constraints imposed by temporal transitive orientations. The latter produces an efficient recognition algorithm for graphs admitting such orientations. Additionally, we extend the temporal transitivity model to temporal graphs having multiple time-labels associated to their edges and claim that the previous results hold in the multilabel setting. Finally, we propose a characterization of temporal comparability graphs via forbidden temporal ordered patterns.

Problem

• Extend Ghouila-Houri's comparability characterization to temporal graphs
• Develop efficient recognition algorithm for temporal transitive orientations
• Characterize temporal comparability graphs using forbidden ordered patterns

Innovation

• Extends Ghouila-Houri's characterization to temporal graphs
• Uses 2-SAT formula to express temporal transitivity constraints
• Provides efficient recognition algorithm for temporal orientations