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Theoretical Computer Science, 2018 We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is motivated by applications in edge casing, as a $k$-gap-planar graph can be drawn crossing-free after introducing ...
Kord Eickmeyer +2 more
exaly +6 more sources
Journal of Combinatorial Theory Series B, 2005 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Angelika Steger, Colin J. H. McDiarmid
exaly +6 more sources
k-L(2, 1)-labelling for planar graphs is NP-complete for k>=4 [PDF]
, 2009 A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common ...
Noble, Steven +8 more
core +6 more sources
The structure and the list 3-dynamic coloring of outer-1-planar graphs [PDF]
Discrete Mathematics & Theoretical Computer Science, 2021 An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge.
Yan Li, Xin Zhang
doaj +1 more source
Improved product structure for graphs on surfaces [PDF]
Discrete Mathematics & Theoretical Computer Science, 2022 Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve
Marc Distel +3 more
doaj +1 more source
Planar Projections of Graphs [PDF]
Discrete Applied Mathematics, 2020 We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices has a representation in $\lceil \sqrt{n/2}+1 \rceil$ planes.
N. R. Aravind, Udit Maniyar
openaire +4 more sources
Good triangulations yield good tours [PDF]
, 2008 Consider the following heuristic for planar Euclidean instances of the Traveling Salesman Problem (TSP): select a subset of the edges which induces a planar graph, and solve either the TSP or its graphical relaxation on that graph. In this paper, we give
Pearson, N, Letchford, A N
core +5 more sources
The nonsolvability by radicals of generic 3-connected planar Laman graphs. [PDF]
, 2007 We show that planar embeddable -connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity.
Power, Stephen C., Owen, J. C.
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On Optimal Beyond-Planar Graphs
Computing in Geometry and Topology, 2023 A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar graphs where every edge is crossed at most k times and RAC graphs where ...
Franz Brandenburg
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Planar Graphs as VPG-Graphs [PDF]
Journal of Graph Algorithms and Applications, 2013 Summary: A graph is \(B_k\)-VPG when it has an intersection representation by paths in a rectangular grid with at most \(k\) bends (turns). It is known that all planar graphs are \(B_3\)-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are \(B_2\)-VPG.
Steven Chaplick, Torsten Ueckerdt
openaire +3 more sources