Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

Beyond Level Planarity

In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of $\textit{Cyclic Level Planarity}$ and $\textit{Torus Level Planarity}$, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the $\textit{Simultaneous Level Planarity}$ of a set of level graphs, wit…

Turning Cliques into Paths to Achieve Planarity

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ …

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 whethe…

拉姆齐理论与同时平面性：顶点完整性与路径宽度

Ramsey Meets Simultaneous Planarity: Vertex Integrity and Pathwidth

该研究针对k个同顶点集图的同时嵌入问题，建立了并图顶点完整性加k参数化的固定参数可处理性，补充了k与并图路径宽度为常数时的NP-难下界，排除了结果向路径/树宽度的推广。

$k$-平面性的结构参数化

Structural Parameterizations of $k$-Planarity

本研究将1-平面性的结构参数化复杂度研究拓展至k≥1的一般情形，给出紧的参数化上下界，证明常数树宽图子类上1-平面性测试仍为NP完全、局部交叉数近似难度更高。

Can an NN model plainly learn planar layouts?

Planar graph drawings tend to be aesthetically pleasing. In this poster we explore a Neural Network's capability of learning various planar graph classes. Additionally, we also investigate the effectiveness of the model in generalizing beyond planarity. We find that the model can outperform conventional techniques for certain graph classes. The model, however, appears to be more susceptible to randomness in the data, and seems to be less robust than expected.