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Planar median graphs and cubesquare-graphs [PDF]
Discrete Applied Mathematics, 2023 Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs.
Carsten R. Seemann +3 more
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The Electronic Journal of Combinatorics, 2019 We say that a graph $H$ is planar unavoidable if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is planar avoidable. That is, $H$ is planar unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from the Four-Color Theorem and a result of
Axenovich, M. +3 more
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Connectivity of Planar Graphs [PDF]
Journal of Graph Algorithms and Applications, 2001 We give here three simple linear time algorithms on planar graphs: a 4-connexity test for maximal planar graphs, an algorithm enumerating the triangles and a 3-connexity test. Although all these problems got already linear-time solutions, the presented algorithms are both simple and efficient. They are based on some new theoretical results.
de Fraysseix, Hubert +1 more
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Untangling a Planar Graph [PDF]
Discrete & Computational Geometry, 2008 A straight-line drawing $δ$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,δ)$ denote the minimum number of vertices that need to be moved to untangle $δ$. We show that shift$(G,δ)$ is NP-hard to compute and to approximate.
Xavier Goaoc +5 more
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A Planarity Criterion for Graphs [PDF]
SIAM Journal on Discrete Mathematics, 2015 It is proved that a connected graph is planar if and only if all its cocycles with at least four edges are “grounded” in the graph. The notion of grounding of this planarity criterion, which is purely combinatorial, stems from the intuitive idea that with planarity, there should be a linear ordering of the edges of a cocycle such that in the two ...
Kosta Dosen, Zoran Petric
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Minimum Cycle Base of Graphs Identified by Two Planar Graphs [PDF]
, 2007 In this paper, we study the minimum cycle base of the planar graphs obtained from two 2-connected planar graphs by identifying an edge (or a cycle) of one graph with the corresponding edge (or cycle) of another, related with map geometries, i.e ...
Han, Ren, Dengju, Ma
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Simplifying Non-Simple Fan-Planar Drawings
Journal of Graph Algorithms and Applications, 2023 A drawing of a graph is fan-planar if the edges intersecting a common edge $a$ share a vertex $A$ on the same side of $a$. More precisely, orienting $a$ arbitrarily and the other edges towards $A$ results in a consistent orientation of the crossings.
Boris Klemz +3 more
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A First Order Logic Definition of Beyond-Planar Graphs
Journal of Graph Algorithms and Applications, 2018 Beyond-planarity is a collective term for classes of graphs that extend the planar graphs and are defined by drawings with restrictions on crossings. Examples are 1-planar, fan-planar, fan-crossing free, and quasi-planar graphs. We define these and other
Franz Brandenburg
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Discrete Applied Mathematics, 2017 A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per ...
Christian Bachmaier +4 more
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Parameterized Complexity of 1-Planarity
Journal of Graph Algorithms and Applications, 2018 We consider the problem of drawing graphs with at most one crossing per edge. These drawings, and the graphs that can be drawn in this way, are called $1$-planar.
Michael Bannister +2 more
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