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 Ï-stream planar with respect to a positive integer window size Ï if there exists a sequence of planar topological drawings Îi of the graphs Gi= (V, ej | i â¤ j < i + Ï) such that the common graph Gi â© = Gi â© Gi+1 is drawn the same in Îi and in Îi+1, for 1 â¤ i < mâÏ. THE STREAM PLANARITY Problem with window size Ï asks whether a given streamed graph is Ï-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 Ï â¥ 2. On the positive side, we provide O(n + Ïm)-time algorithms for (i) the case Ï = 1 and (ii) all values of Ï 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 [GDâ14] for Ï = 1.
DA LOZZO, G., & Rutter, I. (2015). Planarity of streamed graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp.153-166). Springer Verlag.
|Titolo:||Planarity of streamed graphs|
|Data di pubblicazione:||2015|
|Citazione:||DA LOZZO, G., & Rutter, I. (2015). Planarity of streamed graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp.153-166). Springer Verlag.|
|Appare nelle tipologie:||4.1 Contributo in Atti di convegno|