Questions tagged [graph-drawing]

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

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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 ...
L.C. Zhang's user avatar
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1 vote
1 answer
160 views

Who introduced the concept of beyond planar graphs?

The concept of planar graphs seems to be standard (I'm also not sure who first used this term), and recently, beyond planar graphs attract a lot of interest in the field of graph drawing. I know that ...
L.C. Zhang's user avatar
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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 ...
Xin Zhang's user avatar
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5 votes
2 answers
262 views

Enumerating all inequivalent planar embeddings of a planar graph

Graph $G$ can be embedded (or has an embedding) in the space if $G$ can be drawn in the space if $G$ can be drawn in such a way that no two edges cross except at an end-vertex in common. A Graph $G$ ...
L.C. Zhang's user avatar
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3 votes
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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." ...
L.C. Zhang's user avatar
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20 votes
3 answers
1k views

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 ...
L.C. Zhang's user avatar
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7 votes
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148 views

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. &...
L.C. Zhang's user avatar
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1 vote
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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-...
L.C. Zhang's user avatar
  • 1,593
6 votes
1 answer
628 views

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 ...
L.C. Zhang's user avatar
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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, $0$-...
L.C. Zhang's user avatar
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4 votes
1 answer
184 views

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., ...
L.C. Zhang's user avatar
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9 votes
3 answers
402 views

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 ...
W. Paul Liu's user avatar
5 votes
1 answer
207 views

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 ...
L.C. Zhang's user avatar
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4 votes
1 answer
150 views

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 ...
Jukka Kohonen's user avatar
8 votes
1 answer
427 views

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 ...
L.C. Zhang's user avatar
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3 votes
1 answer
132 views

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 ...
L.C. Zhang's user avatar
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1 vote
1 answer
148 views

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 ...
L.C. Zhang's user avatar
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1 vote
1 answer
234 views

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 ...
L.C. Zhang's user avatar
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2 votes
1 answer
115 views

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 ...
Xin Zhang's user avatar
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4 votes
0 answers
128 views

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. ...
Xin Zhang's user avatar
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5 votes
1 answer
256 views

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 ...
Xin Zhang's user avatar
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5 votes
2 answers
244 views

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.
Xin Zhang's user avatar
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3 votes
1 answer
186 views

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 ...
Xin Zhang's user avatar
  • 1,130
7 votes
2 answers
239 views

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?
Xin Zhang's user avatar
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6 votes
2 answers
272 views

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\...
Xin Zhang's user avatar
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1 vote
1 answer
73 views

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 $G$ ...
Hao S's user avatar
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10 votes
2 answers
527 views

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? ...
Szabolcs Horvát's user avatar
5 votes
2 answers
523 views

"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 ...
hbm's user avatar
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7 votes
0 answers
84 views

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 ...
Joseph O'Rourke's user avatar
3 votes
2 answers
123 views

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 ...
Manfred Weis's user avatar
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5 votes
2 answers
470 views

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 ...
Hans-Peter Stricker's user avatar
5 votes
1 answer
476 views

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 ...
Hauke Reddmann's user avatar
5 votes
0 answers
221 views

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 ...
Moritz Firsching's user avatar
4 votes
1 answer
361 views

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 $G$....
user avatar
12 votes
3 answers
1k views

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 ...
Marco Kuhlmann's user avatar
10 votes
2 answers
692 views

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 ...
Joseph O'Rourke's user avatar
1 vote
1 answer
222 views

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 ...
Hooman's user avatar
  • 415
13 votes
1 answer
875 views

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}{...
Jan Kyncl's user avatar
  • 5,886
8 votes
7 answers
1k views

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. ...
Hans-Peter Stricker's user avatar
2 votes
0 answers
245 views

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 ...
Hooman's user avatar
  • 415
3 votes
1 answer
1k views

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 ...
psd's user avatar
  • 266
7 votes
0 answers
335 views

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, ...
David Richter's user avatar
7 votes
2 answers
1k views

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 ...
Suresh Venkat's user avatar