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 e 1 , e 2 , ... , e m 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 Γ i of the graphs G i = (V , { e j | i ≤ j < i + ω }) such that the common graph G ∩ i = G i ∩ G i + 1 is drawn the same in Γ i and in Γ i + 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 + ω m) -time algorithms for (i) the case ω = 1 and (ii) all values of ω provided the backbone graph is 2-connected. Our results improve on the Hanani-Tutte-style algorithm proposed by Schaefer [GD'14] for ω = 1 , which runs in O ((n m) 3) time. [ABSTRACT FROM AUTHOR]