# Questions tagged [planar-graphs]

The planar-graphs tag has no usage guidance.

**4**

votes

**2**answers

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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 ...

**3**

votes

**1**answer

212 views

### Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...

**2**

votes

**0**answers

34 views

### Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...

**3**

votes

**0**answers

38 views

### Reference on generalization of plane graph duality between bonds and simple cycles

Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a ...

**0**

votes

**1**answer

306 views

### How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]

Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...

**10**

votes

**1**answer

202 views

### Orientations of Planar Graphs

Let $G$ be a $2$-edge-connected graph drawn in the plane (such that the edges intersect only at the endpoints). I want to orient
the edges of $G$ such that for each vertex $v$, there are no
three ...

**7**

votes

**0**answers

69 views

### Partitioning the vertices of a graph into induced trees

I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree.
In particular I am ...

**3**

votes

**0**answers

65 views

### Reconstructing plane graphs from degree- and face-sequences

Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...

**9**

votes

**2**answers

206 views

### Graph planarization via rewiring

Let $G$ be a nonplanar graph (undirected) of $n$ nodes and $e$ edges, with
$e \le 3n-6$.
Define a rewiring move as replacing edge $(a,b)$ with edge $(a,c)$.
The result must be a simple graph (no loops,...

**3**

votes

**1**answer

81 views

### A question regarding the all pair shortest paths in weighted planar graphs

What is the time complexity of the fastest known algorithm for the all-pair shortest paths in planar graphs?

**14**

votes

**1**answer

1k views

### Is every graph the center of some other graph?

The center of a graph $G$ is the set of vertices that minimize the largest
distance to vertices in $G$, e.g., in the graph below, that radius is $4$:
Define the ...

**1**

vote

**0**answers

107 views

### Smallest planar graph with two non-homoemorphic plane embeddings?

Apologies for asking a question which probably has a well-known answer:
What is the smallest (not necessarily simple) planar graph with two non-homeomorphic embeddings into the plane?
I am ...

**1**

vote

**1**answer

72 views

### Cholesky factorization of planar graphs

Suppose the sparsity pattern of $A \in \mathbb{R}^{N \times N}$ is a planar graph. Can I use this to bound the complexity of solving
$$
Ax = b
$$
?
In particular, I was hoping to use the planar ...

**4**

votes

**3**answers

455 views

### Does every finite bridgeless cubic planar simple undirected graph admit a 2-factorization with at most two components each of which has even order?

Consider simple bridgeless cubic planar graphs.
Does each such graph admit a 2-factorization with $\leq 2$ components each of which has even order?
If not, does anyone know of an counterexample?
...

**32**

votes

**2**answers

1k views

### A conjecture on planar graphs

I don't know the following is a known result, but it would be very useful to me in my research if it were true.
Conjecture: Let $G$ be a planar graph. The sum
$$
\sum_{\{x,y\} \in E(G)}{\min(deg(x),...

**1**

vote

**1**answer

101 views

### When is an ordering of edges in a graph a planar embedding?

Is there a criterion which decides whether a given rotation system for a graph determines a planar embedding? That is, a lemma of the form:
A graph G = (V, E) is planar if and only if there exists ...

**14**

votes

**2**answers

500 views

### Is there easy proof for triangle-free two-coloring of planar graphs?

By merging two-two color classes, the Four Color Theorem implies that every planar graph can be two-colored such that each color class induces a triangle-free graph.
Is there a simpler proof for this ...

**5**

votes

**1**answer

240 views

### In how many ways can a given planar graph be mapped into the plane?

I feel sure that this question must have been addressed in the literature, but I can't seem to find it - I may be looking in the wrong place.
A graph is planar if it can be drawn on the plane such ...

**3**

votes

**1**answer

369 views

### How non-planar is the Math Genealogy Project graph? [closed]

The Mathematics Genealogy Project keeps an online database of information on as many mathematicians as they can get records for, focusing especially on advisor-student relationships. In this way, they ...

**6**

votes

**1**answer

378 views

### Do planar graphs have an acyclic two-coloring?

A graph has an acyclic two-coloring if its vertices can be colored with two colors such that each color class spans a forest.
Does every planar graph have an acyclic two-coloring?
An affirmative ...