Planarity of streamed graphs

In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A streamed graph is a stream of edges e1, e2,..., em on a vertex set V. A streamed graph is ω-stream planar with respect to a positive integer window size ω if there exists a sequence of planar topological drawings Γi of the graphs Gi= (V, ej | i ≤ j < i + ω) such that the common graph Gi ∩ = Gi ∩ Gi+1 is drawn the same in Γi and in Γi+1, for 1 ≤ i < m−ω. THE STREAM PLANARITY Problem with window size ω asks whether a given streamed graph is ω-stream planar. We also consider a generalization, where there is an additional backbone graph whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the STREAM PLANARITY PROBLEM is NP-complete even when the window size is a constant and that the variant with a backbone graph is NP-complete for all ω ≥ 2. On the positive side, we provide O(n + ωm)-time algorithms for (i) the case ω = 1 and (ii) all values of ω provided the backbone graph consists of one 2-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the Hanani-Tutte-style O((nm)3)-time algorithm proposed by Schaefer [GD’14] for ω = 1.