Planarity of Streamed Graphs
Abstract
In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A $\textit{streamed graph}$ is a stream of edges $e_1,e_2,...,e_m$ on a vertex set $V$. A streamed graph is $\omega$$\textit{stream planar}$ with respect to a positive integer window size $\omega$ if there exists a sequence of planar topological drawings $\Gamma_i$ of the graphs $G_i=(V,\{e_j \mid i\leq j < i+\omega\})$ such that the common graph $G^{i}_\cap=G_i\cap G_{i+1}$ is drawn the same in $\Gamma_i$ and in $\Gamma_{i+1}$, for $1\leq i < m\omega$. The $\textit{Stream Planarity}$ Problem with window size $\omega$ asks whether a given streamed graph is $\omega$stream planar. We also consider a generalization, where there is an additional $\textit{backbone graph}$ whose edges have to be present during each time step. These problems are related to several wellstudied planarity problems. We show that the $\textit{Stream Planarity}$ Problem is NPcomplete even when the window size is a constant and that the variant with a backbone graph is NPcomplete for all $\omega \ge 2$. On the positive side, we provide $O(n+\omega{}m)$time algorithms for (i) the case $\omega = 1$ and (ii) all values of $\omega$ 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 HananiTuttestyle $O((nm)^3)$time algorithm proposed by Schaefer [GD'14] for $\omega=1$.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.07106
 Bibcode:
 2015arXiv150107106D
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 21 pages, 9 figures, extended version of "Planarity of Streamed Graphs" (9th International Conference on Algorithms and Complexity, 2015)