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Advances on Testing C-Planarity of Embedded Flat Clustered Graphs
Lecture Notes in Computer Science, 2014 We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each ...
Fabrizio Frati +2 more
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Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity
Lecture Notes in Computer Science, 2018 The C-Planarity problem asks for a drawing of a clustered graph, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no region-region crossings, and ...
Michael Goodrich +2 more
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A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem
Lecture Notes in Computer Science, 2014 The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions.
Ignaz Rutter
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Planarity for clustered graphs
1995In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study C-planarity of clustered graphs.
Qing-Wen Feng +2 more
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Aromaticity of Planar Boron Clusters Confirmed
Journal of the American Chemical Society, 2005Low-energy boron clusters are characterized by two-dimensional geometry. Aromaticity of these planar boron clusters was established in terms of topological resonance energy (TRE). All planar boron clusters were found to be highly aromatic with large positive TREs even if they have 4n pi-electrons.
Jun-ichi, Aihara +2 more
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The σ delocalization in planar boron clusters
Dalton Transactions, 2009The sigma delocalization plays an important role in the stability of small boron clusters; therefore, it is important to establish under which circumstances this delocalization contributes to the aromaticity of these molecules. In this work, using electron localization function (ELF) calculations, we show that sigma and pi electrons follow different ...
Luis, Rincon +5 more
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Planar graph routing on geographical clusters
Ad Hoc Networks, 2005Geographic routing protocols base their forwarding decisions on the location of the current device, its neighbors, and the packets destination. Early proposed heuristic greedy routing algorithms might fail even if there is a path from source to destination.
Hannes Frey, Daniel Görgen
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Planar soap bubble clusters with a cavity
The European Physical Journal E, 2007We construct local energy-minimizing bubble clusters in the plane that are not simply-connected. Numerical evidence suggests that these minima are not isolated.
C E, Garza-Hume, P, Padilla
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Hierarchical clustering with planar segments as prototypes
Pattern Recognition Letters, 2015Abstract Clustering methods divide a set of observations into groups in such a way that members of the same group are more similar to one another than to the members of the other groups. One of the scientifically well known methods of clustering is the hierarchical agglomerative one.
Jacek M. Leski, Marian Kotas
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How to draw a planar clustered graph
1995In this paper, we introduce and show how to draw a practical graph structure known as clustered graphs. We present an algorithm which produces planar, straight-line, convex drawings of clustered graphs in O(n2.5) time. We also demonstrate an area lower bound and an angle upper bound for straight-line convex drawings of C-planar graphs.
Qing-Wen Feng +2 more
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