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Register a new userVariants of the Segment Number of a Graph
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Added by
Alexander Wolff
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the $existsmathbb{R}$-completeness (which implies the NP-hardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.
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A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.
Let $P subseteq mathbb{R}^2$ be a set of points and $T$ be a spanning tree of $P$. The emph{stabbing number} of $T$ is the maximum number of intersections any line in the plane determines with the edges of $T$. The emph{tree stabbing number} of $P$ is the minimum stabbing number of any spanning tree of $P$. We prove that the tree stabbing number is not a monotone parameter, i.e., there exist point sets $P subsetneq P$ such that treestab{$P$} $>$ treestab{$P$}, answering a question by Eppstein cite[Open Problem~17.5]{eppstein_2018}.
This is the arXiv index for the electronic proceedings of GD 2019, which contains the peer-reviewed and revised accepted papers with an optional appendix. Proceedings (without appendices) are also to be published by Springer in the Lecture Notes in Computer Science series.
This is the arXiv index for the electronic proceedings of GD 2021, which contains the peer-reviewed and revised accepted papers with an optional appendix. Proceedings (without appendices) are also to be published by Springer in the Lecture Notes in Computer Science series.
Proceedings of GD2020: This volume contains the papers presented at GD~2020, the 28th International Symposium on Graph Drawing and Network Visualization, held on September 18-20, 2020 online. Graph drawing is concerned with the geometric representation of graphs and constitutes the algorithmic core of network visualization. Graph drawing and network visualization are motivated by applications where it is crucial to visually analyse and interact with relational datasets. Information about the conference series and past symposia is maintained at http://www.graphdrawing.org. The 2020 edition of the conference was hosted by University Of British Columbia, with Will Evans as chair of the Organizing Committee. A total of 251 participants attended the conference.
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