Streamed Graphs

A streamed graph that is ω-stream planar for ω=2

In this research 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,...,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 Gⁱ_∩=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+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.

References:

• Giordano Da Lozzo, Ignaz Rutter. Planarity of Streamed Graphs. In Proc. 9th International Conference on Algorithms and Complexity (CIAC ’15), Springer-Verlag, Lecture Notes in Computer Science, 2015.
• Giordano Da Lozzo, Ignaz Rutter. Planarity of Streamed Graphs. Technical Report arXiv:1501.07106, Cornell University, 2015.