Planarity of Streamed Graphs

An $omega$-streamed graph that is $\omega$-stream planar for $\omega=2$


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 $e_1,e_2,\dots,e_m$ on a vertex set $V$. A streamed graph is $\omega$-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 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 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 $\mathcal{NP}$-complete even when the window size is a constant and that the variant with a backbone graph is $\mathcal{NP}$-complete 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 Hanani-Tutte-style $O((nm)^3)$-time algorithm proposed by Schaefer [GD’14] for $\omega=1$.

Theoretical Computer Science
Giordano Da Lozzo
Assistant Professor (RTDa)

My research interests are in Algorithm Engineering and Complexity, focused in particular on the theoretical and algorithmic challenges arising from the visualization of graphs.