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Packing Trees into 1-planar Graphs [PDF]
Journal of Graph Algorithms and Applications, 2021 We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and
Felice De Luca +8 more
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The Stub Resolution of 1-planar Graphs
Journal of Graph Algorithms and Applications, 2021 The resolution of a drawing plays a crucial role when defining criteria for its quality. In the past, grid resolution, edge-length resolution, angular resolution and crossing resolution have been investigated.
Michael Kaufmann +5 more
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1-Planarity of Graphs with a Rotation System
Journal of Graph Algorithms and Applications, 2015 A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar
Christopher Auer +3 more
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1-Visibility Representations of 1-Planar Graphs
Journal of Graph Algorithms and Applications, 2014 A 1-visibility representation of a graph displays each vertex as a horizontal vertex-segment, called a bar, and each edge as a vertical edge-segment between the segments of the vertices, such that each edge-segment crosses at most one vertex-segment and ...
Franz Brandenburg
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Linear arboricity of 1-planar graphs
Discussiones Mathematicae Graph TheorySummary: The linear arboricity \(\text{la}(G)\) of a graph \(G\) is the minimum number of linear forests that partition the edges of \(G\). \textit{J. Akiyama} et al. [Networks 11, 69--72 (1981; Zbl 0479.05027)] conjectured that \(\big\lceil\frac{\Delta(G)}{2}\big\rceil\leq \text{la}(G)\leq\big\lceil\frac{\Delta(G)+1}{2}\big\rceil\) for any simple ...
Weifan Wang, Juan Liu, Yiqiao Wang
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Correction to: Outer 1-Planar Graphs [PDF]
Algorithmica, 2021 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Christopher Auer +6 more
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On the Sizes of Bipartite 1-Planar Graphs [PDF]
The Electronic Journal of Combinatorics, 2021 A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite $1$-planar graph with $n$ ($n\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$.
Yuanqiu Huang +2 more
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On Optimal Beyond-Planar Graphs
Computing in Geometry and Topology, 2023 A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar graphs where every edge is crossed at most k times and RAC graphs where ...
Franz Brandenburg
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B0-VPG Representation of AT-free Outerplanar Graphs
Journal of Graph Algorithms and Applications, 2023 A $k$-bend path is a non-self-intersecting polyline in the plane made of at most $k+1$ axis-parallel line segments. B$_{k}$-VPG is the class of graphs which can be represented as intersection graphs of $k$-bend paths in the same plane. In this paper,
Sparsh Jain +2 more
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Drawing outer-1-planar graphs revisited
Journal of Graph Algorithms and Applications, 2022 In a recent article (Auer et al., Algorithmica 2016) it was claimed that every outer-1-planar graph has a planar visibility representation of area $O(n\log n)$.
Therese Biedl
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