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

Abstract

An orientation of a 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 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 due to Ghouila-Houri [Ghouila-Houri, 1962] states that any static graph admitting a quasi-transitive orientation also admits a transitive orientation. In a seminal work [Mertzios et al., 2025], 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 structural theorem that will allow us to express by a 2-SAT formula all the constraints imposed on a temporal graph for it to admit a temporal transitive orientation. The latter produces an efficient recognition algorithm for graphs admitting such orientations, that we will call comparability temporal graphs. Inspired by the lexicographic strategy presented by Hell and Huang in [Hell and Huang, 1995] to transitively orient static graphs, we then propose an algorithm for constructing a temporal transitive orientation of a YES instance. This algorithm is straightforward and has a running-time complexity of O(nm + min{kn,m²}), with n, m and k being respectively the number of vertices, edges and monolabel triangles, i.e., triangles having the same unique time-label on their edges, in the temporal graph. This represents an improvement compared to the algorithm presented in [Mertzios et al., 2025]. 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 comparability temporal graphs by forbidden temporal ordered patterns.