Questions tagged [graph-drawing]
Problems related to graph drawing such as crossing numbers, layout designs, and intersection graphs.
16 questions with no upvoted or accepted answers
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Why is the crossing number of Tutte 12-cage 170?
From the Wikipedia entry on Tutte 12-cage , it is stated that the crossing number of Tutte 12-cage is 170, but the cited references do not seem to provide sufficient explanation for this.
Exoo, G. &...
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Universal point sets for 1-plane graphs
It is a notorious open problem to find a smallest set of $N$ points that
permit any $n$-vertex planar graph to be drawn in the plane without
crossings, using only those $N$ points as vertices, and ...
- 151k
7
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3-dimensional Cayley graph
I would like to see Cayley graphs drawn in 3-dimensional Euclidean space such that the vertices are represented by points and various shadows display the actions of the generators.
For example, ...
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5
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"separators" for nonplanar graphs embedded in the plane
Given a nonplanar graph $G$ drawn in the plane with crossings.
Does there exist a small ($o(|V(G)|$) subset $S$ of edges of $G$ such that after the removal of all edges that intersect or share an ...
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5
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Minimal algebraic degree of symmetric unit distance embedding of Heawood graph
I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...
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4
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What is the crossing number of dodecahedron with a copy of $K_5$ inside each face
Suppose we are given a regular dodecahedron. Then we add five crossed edges inside each of its faces (actually, inside each face it is a copy of $K_5$). It is clear that this drawing has 60 crossings. ...
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Is there a more intuitive proof that a 1-planar graph with minimum degree 7 contains a $K_4$?
In the following paper, Hudák Dávid, and Tomáš Madaras give the following Theorem 1.1.
Hudák, Dávid, and Tomáš Madaras. "On local properties of 1-planar graphs with high minimum degree." ...
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Is the chromatic number of every 7-connected 1-planar graph at most 5?
1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors. Later, the precise number of colors needed to color these graphs, in the worst case, ...
- 1,851
2
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What is the range of connectivity for maximal IC-planar graphs?
A graph is IC-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 no common end vertex. A graph $G$ is maximal in a graph ...
- 1,851
2
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A primal-dual (double) circle packing (coin graph) question
I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem).
Q1- But I am not sure whether ...
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What lower bounds are known for pair crossing number and related questions in multigraphs?
So in terms of crossing number https://arxiv.org/pdf/1808.10480 gives a lower bound of $O(e^{2.5}/n^{1.5})$ for multigraphs with no face of length 2 with no node contained inside.
What do we know ...
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Problem related to crossing number
Let $G$ be a graph embedded in the plane (with crossings). For $ F \subset E(G) $, denote by $c(F)$ the set of edges of $G$ that cross some edge in $F$.
Denote $\delta(v)$ the set of edges with one ...
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Is there a variant of the crossing lemma for multigraphs with arbitrary embedding?
Suppose $G$ is a graph embedded in the plane with $m=|E(G)|$ edges and $n=|V(G)|$ vertices.
Suppose $\operatorname{sim}(G)$, the simplification of $G$ contains $ m' \gg 3n $ edges.
Call the set of ...
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1
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A confusion about the proof of maximal 1-plane graph being $2$-connected
It is well known that every maximal planar graph with at least 4 vertices is 3-connected. But for maximal 1-planar graphs we cannot ensure the high connectivity. (See is-there-any-maximal-1-planar-or-...
- 1,851
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Does contracting a non-crossing edge of a $k$-plane graph change the $k$-planarity?
A graph is $k$-planar if it can be drawn on the plane such that each edge is
crossed at most $k$ times. A graph together with a $k$-planar drawing is a $k$-plane
graph. Hence, by definition, $0$-...
- 1,851
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Is there is a constant $c$ such that toroidal graphs are minor-$c$-colorable?
A toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
A minor of graph G is a graph obtained from G by ...
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