# Questions tagged [graph-drawing]

Problems related to graph drawing such as crossing numbers, layout designs, and intersection graphs.

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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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### Can a 3-regular non-1-planar graph be constructed?

A $1$-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing.
I used nauty to generate all 3-regular graphs up to ...

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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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### 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-...

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### How to construct a 5-regular 1-planar bipartite graph?

A graph is 1-planar if it can be drawn on the plane such that each edge is
crossed at most once.
Let $G$ be a 1-planar bipartite graph with $n~(n > 4)$ vertices and $m$ edges. Karpov [1] showed ...

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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, $0$-...

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### Find all 2-planar drawings of $K_6$ and $K_7$

A $k$-planar graph is a graph which can be embedded with at most $k$ crossings per
edge.
It is proved that a complete graph $K_n$ is 2-planar if and only if $n\le 7$.
Angelini P., Bekos M. A., ...

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### Is it possible that every edge in a 1-planar drawing with minimum number of crossings is crossed?

A graph is 1-planar is it has drawing in the plane so that each edge is crossed at most once. Here we also assume the drawing satisfies (1) no edge is self-crossed; (2) no two adjacent edges are ...

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### Is the crossing number of the line graph of $K_5$ determined?

The line graph of an undirected graph $G$ is another graph $L(G)$ that represents the adjacencies between edges of $G$. $L(G)$ is constructed in the following way: for each edge in $G$, make a vertex ...

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### Which Boolean lattices have a left-to-right symmetric drawing?

This question is inspired by a similar MSE question about partition lattices.
Question: Which finite Boolean lattices have a symmetric drawing on the 2D plane?
By a symmetric drawing of a lattice, I ...

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### Find all Non-isomorphic good drawings of $K_{3,3}$？

Sometimes I look at all non-isomorphic good drawings of graphs on a plane or sphere.
Good drawing means that no edge crosses itself, no two edges cross more than once, and no two edges incident with ...

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### Constructing a 1-planar graph that has no rectilinear drawing

A 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge.
1 planar graph
I read the ...

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### Can Tutte embedding be guaranteed that each face is convex?

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a ...

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### Abnormal toroidal drawing of graph

1. Some background knowledge
Definition. A torus, informally, is the doughnut-shaped surface that we get by taking a square made out of some arbitrarily-stretchy material and gluing together opposite ...

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### The density of a tripartite 1-planar graph

1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $n$-vertex 1-planar graph is at most $4n-8$, and the ...

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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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### What is the crossing number of cube with a pair of crossing edges inside each face

Suppose we are given a cube and we add a pair of crossing edges inside each of its faces. It is clear that this drawing has 6 crossings. My question is whether such a graph has crossing number 6? How ...

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### Finding a special plane graph with some requirements on the faces

Is there a plane graph such that
(1) the outer face has degree 3, i.e, is a triangle,
(2) every inner face has degree 5, and
(3) any two degree 5 faces share at most one commong edge.

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### Looking for examples showing that the crossing number may not be realized by the drawings with local crossing number

The crossing number $cr(G)$ of a graph $G$ is the lowest number of edge crossings of a plane drawing of the graph $G$. The local crossing number of a drawing of a graph is the largest number of ...

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### There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....?

Is there any characterization on the set of integers $n$ such that there is a 3-connected 5-regular simple $n$-vertex planar graph?

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### Is there any maximal 1-planar or 2-planar graph that is not 3-connected

A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. A $k$-planar graph $G$ is maximal if $G+uv$ is not $k$-planar for any non-adjacent vertices $u,v\...

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### Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph, what is the smallest number of "curves"?

Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph $G$, what is the smallest number of "curves" in the plane drawn from $u$ to $v$ such that no $u$--$v$ path in $G$ ...

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### Method to draw 3-connected planar graph on a sphere

The Tutte embedding is a way to create a "nice" drawing of a 3-connected planar graph in the plane, after having chosen an outer face.
Is there a similar method to draw such a graph on a sphere? ...

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### "Locally Nonplanar" graph

A 2-connected $3$-connected graph $G$ is "Almost Planar" Locally Nonplanar if it has a a $2$-connected spanning subgraph $H$ and an embedding in the plane such that $H$ is planar in this embedding and ...

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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 ...

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### Number Associated with Straight-line Drawings of Hamiltonian Graphs

Is there anything known about the maximum number of simple-polygonal Hamilton cycles that a straight-line drawing of a Hamiltonian graph can have?
Put differently, if the vertices of a Hamilton ...

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### Another graph characteristic

This question concerns a method of drawing graphs and a graph characteristic about which I want to learn more.
Consider a connected directed graph with at least one node with in-degree 0 and one node ...

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### Can all crossings in a graph be moved to one point?

Consider a graph $G$ with at least two unavoidable crossings, say, the disjoint union of two copies of $K_5$. Can such a graph always be drawn so that there is only one singular point (where all ...

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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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### Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing.
The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of $G$....

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### Is every graph an edge-crossing graph?

Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph ...

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### Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the
plane so that each arc is crossed at most once by another arc.
A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times.
Planar graphs are ...

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1
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### Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...

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### Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor \left\lfloor\frac{n-1}{...

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### Visualizing polyhedra from their 1-skeletons

Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here.
...

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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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### Graph drawing: unrooted undirected tree graphs with specified edge lengths

Has Joseph Felsenstein's equal daylight layout been analyzed by the graph drawing community? The following description is taken from his
drawtree
documentation (Wayback Machine):
"This ...

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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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### Planar layouts of bipartite graphs

Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar.
It would be very convenient if there was a ...