Results 21 to 30 of about 433,943 (319)
1-Visibility Representations of 1-Planar Graphs
Journal of Graph Algorithms and Applications, 2014 A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices; beyond that segments do not intersect.
Franz J. Brandenburg
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Three classes of 1-planar graphs [PDF]
, 2014 A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. In this paper we decompose the set of all 1-planar graphs into three classes C0, C1 and C2 with respect to the types of crossings ...
Július Czap, Peter Šugerek
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On total colorings of 1-planar graphs [PDF]
Journal of Combinatorial Optimization, 2013 A graph is $$1$$1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree at least 13.
Xin Zhang, Jianfeng Hou, G. Liu
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Non-1-Planarity of Lexicographic Products of Graphs
Discussiones Mathematicae Graph Theory, 2021 In this paper, we show the non-1-planarity of the lexicographic product of a theta graph and K2. This result completes the proof of the conjecture that a graph G ◦ K2 is 1-planar if and only if G has no edge belonging to two cycles.
Matsumoto Naoki, Suzuki Yusuke
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The Book Thickness of 1-Planar Graphs is Constant [PDF]
Algorithmica, 2015 In a book embedding, the vertices of a graph are placed on the “spine” of a book and the edges are assigned to “pages”, so that edges on the same page do not cross.
M. Bekos +3 more
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Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends [PDF]
International Symposium Graph Drawing and Network Visualization, 2018 We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively.
S. Chaplick +3 more
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On the Density of Maximal 1-Planar Graphs [PDF]
International Symposium Graph Drawing and Network Visualization, 2012 A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity. Maximal 1-planar graphs have at most 4n−8 edges. We show that there are sparse maximal 1-planar graphs with only $\frac{45}{17} n + \mathcal{O}(1)$ edges.
F. Brandenburg +5 more
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On local properties of 1-planar graphs with high minimum degree
Ars Math. Contemp., 2011 A graph is called 1-planar if there exists its drawing in the plane such that each edge contains at most one crossing. We prove that each 1-planar graph of minimum degree 7 contains a pair of adjacent vertices of degree 7 as well as several small graphs ...
David E. Hudak, Tomáš Madaras
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On the lightness of chordal 4-cycle in 1-planar graphs with high minimum degree
Ars Math. Contemp., 2013 A graph G is 1 -planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. The family of 1-planar graphs with minimum vertex degree at least δ and minimum edge degree at least ɛ is denoted by P δ 1 ( ɛ ) .
Xin Zhang, Guizhen Liu
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Fáry's Theorem for 1-Planar Graphs
International Computing and Combinatorics Conference, 2012 A plane graph is a graph embedded in a plane without edge crossings. Fary’s theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fary’s theorem to a class of non-planar graphs.
Seok-Hee Hong +3 more
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