Planarity of Streamed Graphs

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 $ω$-$\textit{stream planar}$ with respect to a positive integer window size $ω$ if there exists a sequence of planar topological drawings $Γ_i$ of the graphs $G_i=(V,\{e_j \mid i\leq j < i+ω\})$ such that the common graph $G^{i}_\cap=G_i\cap G_{i+1}$ is drawn the same in $Γ_i$ and in $Γ_{i+1}$, for $1\leq i < m-ω$. The $\textit{Stream Planarity}$ 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 $\textit{backbone graph}$ whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the $\textit{Stream Planarity}$ 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 $ω\ge 2$. 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 $ω=1$.