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The Basis Number of 1-Planar Graphs

Annals of Combinatorics
Let $B$ 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, Therese Biedl, Prosenjit Bose, Anil Maheshwari, Babak Miraftab   +4 more
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On RAC drawings of 1-planar graphs

Theoretical Computer Science, 2017
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Michael A. Bekos, Walter Didimo, Giuseppe Liotta, Saeed Mehrabi 0001, Fabrizio Montecchiani   +4 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, Franz J. Brandenburg, Stephen G. Kobourov   +2 more
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Joins of 1-planar graphs [PDF]

Acta Mathematica Sinica, English Series, 2014
A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is subgraph-majorized (that is, both $G$ and $H$ are subgraphs of graphs of the major pair) by one of pairs $[C_3 \cup C_3,
Czap, Július, Hudák, Dávid, Madaras, Tomáš   +2 more
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Fine-grained complexity of coloring unit disks and balls

Journal of Computational Geometry, 2018
On planar graphs, many classic algorithmic problems enjoy a certain "square root phenomenon" and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set, 3-Coloring, Hamiltonian Cycle ...
Csaba Biró, Édouard Bonnet, Dániel Marx, Tillmann Miltzow, Paweł Rzążewski   +4 more
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Minimal non-1-planar graphs

Discrete Mathematics, 2008
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On the discrete Fréchet distance in a graph

Journal of Computational Geometry
The Fréchet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and ...
Anne Driemel, Ivor van der Hoog, Eva Rotenberg   +2 more
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Additive List Coloring of Planar Graphs with Given Girth

Discussiones Mathematicae Graph Theory, 2020
An additive coloring of a graph G is a labeling of the vertices of G from {1, 2, . . . , k} such that two adjacent vertices have distinct sums of labels on their neighbors.
Brandt Axel, Jahanbekam Sogol, White Jennifer   +2 more
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Linear arboricity of 1-planar graphs

Discussiones Mathematicae Graph Theory
Summary: 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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On Independent Domination in Planar Cubic Graphs

Discussiones Mathematicae Graph Theory, 2019
A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S.
Abrishami Gholamreza, Henning Michael A., Rahbarnia Freydoon   +2 more
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