Planarity of streamed graphs

Abstract

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 G∩i=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 is 2-connected. Our results improve on the Hanani-Tutte-style algorithm proposed by Schaefer [GD'14] for ω=1, which runs in O((nm)3) time.