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

TL;DR

This work extends the theory of comparability graphs from static to temporal graphs by introducing a temporal analogue of Ghouila-Houri’s characterization. Central to the approach is the Almost-TTO (ATTO) framework and the augmented implication digraph, which together yield a structured, polynomial-time method to decide whether a temporal graph admits a temporal transitive orientation ($TTO$) and to construct one when possible. The authors also generalize to multilabel temporal graphs, providing equivalent characterizations and efficient algorithms, and they establish forbidden-ordered-pattern frameworks that connect orientation properties to pattern avoidance under vertex orders. Together, these results deliver a unified, algorithmically tractable theory for temporal comparability and transitivity, with extensions to stronger variants and a suite of open directions on modular decompositions, relabeling, and permutation-based temporal graphs.

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.

Figures (8)

• Figure 1: $(i)$ Comparability graph $K_4$ as underlying static graph of a non T-comparability temporal graph. $(ii)$ Non-comparability graphs as underlying static graphs of T-comparability temporal graphs.
• Figure 2: A non T-comparability graph admitting a QTTO
• Figure 3: Correlated monolabel triangle $(a,b,c,d)$
• Figure 4: $TTO$ temporal ordered pattern
• Figure 5: Strict-TTO forbidden temporal ordered patterns

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