Questions tagged [planar-graphs]

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What is the standard definition of dual of disconnected planar graph when underlying graph derives 'product structure' over connected graphs?

Dual graph of a plane graph has a standard definition https://en.wikipedia.org/wiki/Dual_graph and an edgeless graph on $n$ vertices is planar. What is the standard dual graph of such a graph? Update ...
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1answer
110 views

Implementation of Koebe–Andreev–Thurston circle packing?

The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the ...
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42 views

Planarity of a subgraph

Given a complete symmetric graph $G(V,E)$ with real-valued edgeweights and assume that every induced $k_4$ that is induced by a quadruplet from $v$'s vertices has a unique perfect matching of maximal ...
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124 views

Does there exist 2-planar graph with chromatic number 8 or 9 or 10

A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $|E(G)|\le 5(|V(G)|-2)$. This implies that ...
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65 views

crossing number and thickness of a simple graph $G$

Crossing number$($cr$)$: The crossing number of a simple graph is the minimum number of crossings that can occur when this graph is drawn in the plane where no three arcs representing edges are ...
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0answers
232 views

Reference for results about planar graphs

A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
4
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0answers
66 views

Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 14?

Consider a planar graph where every vertex is incident to at least 3 edges, and assign to each edge a weight equal to the sum of the degrees of its endpoints. If not, what is the smallest n so that ...
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181 views

The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each ...
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50 views

Planar graphs with perfect matching count in linear time?

We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
2
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0answers
72 views

Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
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1answer
167 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? ...
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2answers
118 views

Does any long path in a planar graph contain one of O(n) k-tuple of vertices?

My question is a bit related to both the container method and shallow cell complexity. Let's start with that the number of length $\ell$ paths (where $\ell$ denotes the number of vertices of the path!)...
4
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2answers
303 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
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1answer
245 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\...
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43 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 ...
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80 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 ...
10
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1answer
230 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 ...
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109 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 ...
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85 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 ...
9
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2answers
224 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,...
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1answer
100 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?
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150 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 ...
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3answers
472 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? ...
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1answer
329 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 ...
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1answer
79 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 ...
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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),...
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1answer
120 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 ...
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1answer
258 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 ...
14
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2answers
627 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 ...
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1answer
389 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 ...
7
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1answer
392 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 ...
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101 views

Determine sub-polygon from line segments with known member connectivity

Test Polygon: Consider the following polygon as attached. Let the known parameter be as follows: •Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members. •...
14
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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 ...