Brief Announcement: Revisiting the Realizability of Periodic Temporal Graphs with Bounded Stretch

Abstract

In this work, we revisit Stretched Periodic Temporal Graph Realization (STGR) which was recently introduced by Mertzios et al. [MFCS 2025]. Here, the input consists of an undirected graph G = (V,E), a period Δ, and a rational number α ≥ 1, and the question is, whether there is a labeling λ: E → [0,Δ-1], such that the stretch is at most α in the Δ-periodic temporal graph (G,λ), that is, the temporal graph, where for each c ∈ ℕ and each edge e, e appears at time c⋅ Δ + i if and only if λ(e) = i. The stretch of (G,λ) is the maximum stretch between any vertex pair (u,v) in (G,λ), where the stretch of a vertex pair (u,v) is defined as the duration of a fastest temporal path from u to v in (G,λ) divided by the distance between these vertices in the underlying graph. We complete the complexity picture for STGR with respect to Δ by investigating the open case of Δ = 2. It turns out that STGR is NP-hard for each Δ > 1. Moreover, we also answer the open question by Mertzios et al. on whether there are graphs for which the smallest possible stretch is larger than (Δ+1)/2. We show not only that such graphs exist, but also that it remains NP-hard to decide whether the optimal stretch is at most (Δ+1)/2. Our hardness results for Δ = 2 also imply hardness for Δ = 2 for the Fastest Periodic Temporal Graph Realization problem that was introduced by Klobas et al. [TCS 2025]. Finally, we show the existence of classes of graphs with small and large stretch.