Use for questions about embeddings of graphs in surfaces of genus greater than 0. For embeddings of graphs in planes, spheres, and other simply-connected spaces, use [planar-graphs].
Questions tagged [graph-embeddings]
29 questions
17
votes
1 answer
Genus of the graph $K_{4,2,2,2}$.
What is the genus of the complete $4-$partite graph $K_{4,2,2,2}$?
What i know: Since $K_{4,4,2}$ is a subgraph of $K_{4,2,2,2}$, and genus of $K_{4,4,2}$ is 2, $K_{4,2,2,2}$ has genus greater than or equal to 2. Also $K_{4,2,2,2}$ is a subgraph of…
bor
- 1,063
5
votes
1 answer
How can I decide if this regular graph is toroidal?
I am considering the following graph $G$ with $10$ vertices and $30$ edges:
Vertices of $G$ are the ten two-element subsets of $\{A,B,C,D,E\}$
Vertices $u$ and $v$ are adjacent if $$|v\cap u| = 1.$$
So for example the vertex $AB$ is adjacent to…
MJD
- 67,568
- 43
- 308
- 617
4
votes
1 answer
On the circular embedding of Nguyen Huy Xuong's graph
The following is a conjecture in graph theory.
Circular Embedding Conjecture. Every 2-connected graph has a circular embedding in some surface.
An embedding of a graph in a surface is circular or strong if every face of the embedding is bounded by…
Alma Arjuna
- 6,521
4
votes
1 answer
What is the lowest dimension for topological or isometric embedding of Y^n into euclidean space?
Let $Y$ denote the one-point union of three unit intervals. Since $Y$ embeds isometrically in the plane, it follows that the nth cartesian power $Y^n$ embeds isometrically — and also topologically — in $\mathbb R^{2n}$.
The metric on $Y^n$ is the…
Dan Asimov
- 1,177
3
votes
1 answer
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 endpoint in $v$.
For a node $v \in G $, denote $…
Hao S
- 504
3
votes
2 answers
Is it true if a face of a graph is not homeomorphic to an open disk, then we may find a noncontractible curve contained in the face?
Suppose we have a graph embedded on a surface $Q$ and one face $F$ of the graph is not homeomorphic to an open disk. Does there exist a closed (smooth nonselfinteresecting) curve $g$ contained in $F$ such that $g$ does not divide $Q$ into two…
Hao S
- 504
3
votes
1 answer
Can we draw the dual graph with straight lines?
I've been investigating this problem:
Given some straight line planar embedding of a simple connected graph with a simple dual graph, does there exist a straight line planar embedding of that dual graph with each vertex lying in its corresponding…
Tbw
- 1,015
- 7
- 13
2
votes
1 answer
Conditions for graph embedded into hyperbolic space to be acyclic
I am thinking about the following problem: Let $G$ be a graph embedded into hyperbolic space $\mathbb{H}^d$, i.e., the vertex set $V(G)$ is a subset of $\mathbb{H}^d$ and the edges are geodesic line segments connecting vertices.
Let us call such a…
tikon
- 83
- 6
2
votes
1 answer
A theorem about graph embeddings
I'm trying to learn about graph embeddings and came across a procedure called the "capping operation", as shown in Minimal Imbeddings and the Genus of a Graph and also proved in Monotone Transformations of Two-Dimensional Manifolds. Defining the…
ARYAAAAAN
- 1,709
2
votes
1 answer
Genus of a graph consisting of two faces homeomorphic to open disks
Suppose the graph $G$ is embedded in a surface $Q$ such that there are two faces $F_1,F_2$ of the embedding, each homeomorphic to the open disk, such that each node of $G$ lies on $F_1$ or $F_2$.
Is the genus of $G$ necessarily a constant?
Hao S
- 504
2
votes
1 answer
Relationship between the Crossing Number & the Genus of a graph
The Crossing Number of a graph is the minimum value of the crossing point among all drawings...
on the other hand, using Euler's formula, we know that a graph is embeddable in a space with a sufficiently large genus.
Since we can consider each hole…
MR_BD
- 6,307
1
vote
2 answers
Counterexample proving that the family of 1-planar graphs is not minor-closed
Recently I came across an exercise where we were asked to prove that the family of 1-planar graphs is not minor-closed. In this article they give an example (figure 2) and say that after contracting edge $(2,4)$, the obtained graph is not…
hajduzs
- 13
1
vote
1 answer
A 3D cube plus a diagonal edge is not embeddable in any hypercube
How do I show that a graph created by adding a single edge between antipodal vertices of a 3-dimensional cube is not hypercube embeddable?
Section 2.1 in this paper states that
A connected graph $G$ can be embedded into $Q_n$ (hyper cube of $n$…
Ning Bao
- 75
1
vote
0 answers
Dipping into sets of parallel edges in graph drawings
Given a multigraph $G$ embedded in the plane call a maximal set of parallel edges between $u,v$ such that only one of the induced faces (a face is a connected region of $\mathbb{R}^2 \backslash G $) contains nodes besides $u$ or $v$ a topologically…
Hao S
- 504
1
vote
0 answers
Find the best embeddding for this gene/lipid graph
I want to find a 'nice' drawing of the lipids and genes in my database. Lipids belong to one one of several classes, while genes belong to one of several regions. Each gene/lipid pair has an associated strength value, which indicates how closely…
TRP
- 163