Results 51 to 60 of about 17,567 (164)
The Basis Number of 1-Planar Graphs
Annals of CombinatoricsLet $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$, denoted by $b(G)$, is the smallest integer such that $G$ has a $k$-basis. A graph is called 1-planar (resp.
Saman Bazargani +4 more
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On properties of maximal 1-planar graphs
Discussiones Mathematicae Graph Theory, 2012 Department of Mathematics, Faculty of ScienceNiigata University8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japane-mail: y-suzuki@math.sc.niigata-u.ac.jpAbstractA graph is called 1-planar if there exists a drawing in the plane so thateach edge contains at most one crossing.
Dávid Hudák +2 more
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On Alliance Partitions and Bisection Width for Planar Graphs
Journal of Graph Algorithms and Applications, 2013 An alliance in a graph is a set of vertices (allies) such that each vertex in the alliance has at least as many allies (counting the vertex itself) as non-allies in its neighborhood of the graph. We show how to construct infinitely many non-trivial
Martin Olsen, Morten Revsbaek
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Equitable coloring in 1-planar graphs
Discrete Mathematics9 ...
Daniel W. Cranston, Reem Mahmoud
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The Book Thickness of 1-Planar Graphs is Constant [PDF]
Algorithmica, 2016 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. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant ...
Michael A. Bekos +3 more
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Structural Parameterizations of $k$-Planarity
Journal of Graph Algorithms and ApplicationsThe concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times.
Tatsuya Gima +2 more
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$1$-string $B_2$-VPG representation of planar graphs
Journal of Computational Geometry, 2016 In this paper, we prove that every planar graph has a 1-string $B_2$-VPG representation—a string representation using paths in a rectangular grid that contain at most two bends.
Therese Biedl, Martin Derka
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The strong chromatic index of 1-planar graphs [PDF]
Discrete Mathematics & Theoretical Computer ScienceThe chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors.
Yiqiao Wang +3 more
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On the Book Thickness of 1-Planar Graphs
CoRR, 2015 In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page do not cross each other. The minimum number of pages in which a graph can be embedded is called the book-thickness
Md. Jawaherul Alam +2 more
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3D Visibility Representations of 1-planar Graphs [PDF]
, 2018 We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles.
Patrizio Angelini +3 more
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