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 omega-stream planar with respect to a positive integer window size. if there exists a sequence of planar topological drawings Gamma(i) of the graphs G(i) = (V, {e(j) vertical bar i <= j < i + omega}) such that the common graph G(boolean AND)(i) = G(i) boolean AND G(i+1) is drawn the same in Gamma(i) and in Gamma(i+1), for 1 <= i < m - omega. The STREAM PLANARITY Problem with window size omega asks whether a given streamed graph is.-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 omega >= 2. On the positive side, we provide O(n + omega m)-time algorithms for (i) the case omega = 1 and (ii) all values of. 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 omega = 1.