Questions tagged [planar-graphs]
The planar-graphs tag has no usage guidance.
4
votes
2answers
240 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 ...
3
votes
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
2answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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 ...
