Questions tagged [planar-graphs]
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Practical computation time, counting spanning trees and selecting spanning trees uniformly at random
I am doing a project in applied math, which involves counting spanning trees and selecting spanning trees uniformly at random for near-maximal planar graphs with ~430 vertices, as part of a larger ...
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1answer
120 views
Need help to come up with definitive proofs with regard to Planar Graphs
I was working through a few problem sets and came across this question
Suppose there is a connected planar simple graph $G$ with $v$ vertices such that all its regions are triangles (a cycle ...
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1answer
48 views
How would I algorithmically “stretch” polygons on a plane by re-scaling the distances between interior points?
I've been thinking about a computational problem and could use some guidance for how to go about developing an algorithm to solve it.
On a Euclidean plane, I have a polygon A, a set of points A* ...
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25 views
Simple proof that finding a combinatorial map of a planar graph given as an incidence matrix can be done in polynomial time?
Suppose that I have a graph $G = (V,E)$, given as an incidence matrix of edges and vertices. Suppose that $G$ is planar, that is, it can be embedded in the plane without edge crossings.
I would like ...
3
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1answer
63 views
Why is the graph inside Graham Scan always planar
One of the ways to prove that Graham Scan constructs convex hull in linear time is using planarity of the graph obtained by running the algorithm. This graph is always planar, so according to Euler's ...
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1answer
63 views
Planar embeddings of planar graph
Let $G=(V,E)$ be a planar graph, only specified by its set of vertices and edges. Suppose $|V|=n$. According to Fary's theorem, there exists a planar embedding of $G$ with straight line segments. Then ...
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1answer
60 views
Is every planar graph a possible dual graph of a voronoi diagram?
My question is: Given a planar graph defined by its adjacency matrix. Can I always find a set of points, so that the dual graph of the voronoi diagram of that set of points is the same as the planar ...
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30 views
Tight condition making unit-disk-graphs planar
Here's a nice property about unit-disk-graphs :
Suppose $V\subseteq\mathbb{R}^2$ is a finite set of points in the plane. Build the graph $G_V=(V,E)$ such that $(v,v')\in E$ iff $d(v,v')\le2$, where ...
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1answer
43 views
Where is the second face in a graph with 3 nodes?
I understand that to work out the number of faces of a connected planar graph, you use Euler's formula F = A - N + 2, where A is the number of arcs and N is the number of nodes.
For a triangle node (...
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0answers
76 views
Is there an optimization problem on planar graphs which is APX-hard ?
I'm looking for a optimization problem on planar graphs which is APX-hard, which means that it doesn't admit a PTAS (approximation scheme). It would be even better is the difficulty of the problem ...
3
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107 views
Convert DAG whose transitive reduction is non-planar to a planar DAG with same transitive closure
For any non-planar DAG, we can compute the transitive reduction, and sometimes it will be planar (e.g. $K_5$).
However, sometimes the transitive reduction is also non-planar. For example, $K_{3, 3}$'...
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1answer
95 views
Edge removal to convert a non-planar DAG to a planar DAG while maintaining reachability?
Is there an algorithm that removes edges to convert a non-planar DAG to a planar DAG while maintaining reachability? For example, the graph $G$ below is non-planar:
but, by removing certain edges to ...
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1answer
70 views
Show that graph is planar or not?
Show that the following graph is planar or not.
My first assumption is that this graph is not planar, but could not find a reasonable prove (except saying that I tried drawing it in different ways in ...
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0answers
112 views
Counting the number of connected components in a dynamic plane graph
I'm working on the following problem: let $G = (V, E)$ be a connected, planar graph. Our goal is to find a $d$-partition of $G$, $P = \{V_1, \ldots, V_d\}$, such that $G[V_i]$ is connected, and $\min_{...
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1answer
49 views
Infinite sequence of graphs
http://jgaa.info/accepted/2011/HasheminezhadMcKayReeves2011.15.3.pdf
Hello I stumbled upon this paper. On page $4/20$ the figure shows an infinite sequence of $5-$regular, connected planar graphs. ...
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1answer
90 views
Planar graphs of bounded degree: mixing time = cover time?
For many planar graphs of bounded degree (binary tree, lattice, cycle) the (1/4)-mixing time and the cover time are equal, up to log-factors. Is this always the case?
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1answer
47 views
Planar Embedding with Some Nodes Constrained
I've read about basic planar-graph embedding and about embedding a planar graph onto a set of fixed points, but I was wondering how one might constrain the locations of some nodes—perhaps to a set of ...
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1answer
436 views
Why graph planarity is important
What is the reason to study planar graphs and algorithms on such graphs (as well as algorithms allowing to check a graph's planarity)? Where in industry this knowlege is needed?
I know that planarity ...
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0answers
174 views
FPT: Dominating Set on Planar Graphs (average degree is known)
I'm given an instance of a planar graph and should construct a FPT algorithm for dominating set. The task looks like this:
Dominating Set on Planar Graphs
Instance: A planar graph G and an
integer ...
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0answers
24 views
Planar cover of a rigid and non-planar graph
Let $G$ be a graph which is rigid and non-planar (e.g. $K_{3,3}$). Is it possible that $G$ has a planar cover?
Are there any studies on this topic?
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1answer
1k views
Treewidth of k x k square grid graphs
According to some slides I found on google, the treewidth of any $k \times k$ square grid graph $G$ is $tw(G) = k$. I just started researching about treewidth and tree decomposition, and for the most ...
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1answer
100 views
Partitioning an undirected, unweighted, square planar graph paths that join certain pairs of nodes
I am trying to find a way to efficiently solve a puzzle that I play a lot by turning it into a graph partitioning problem (which is basically is in its actual form). I know that generally, graph ...
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1answer
739 views
Converting a non-planar graph to planar
Suppose that we have a non-planar graph $G$ which is undirected and connected. Our aim is to remove a set of edges and/or a set of vertices and convert make $G$ planar while keeping the connectedness.
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44 views
Complete set of basic circuits for McLane's Theorem
I was assigned a project in which i had to implement some algorithms concerning graphs. The last one is the one described in the title. I have to make an algorithm that uses McLane's theorem (https://...
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Maximum Weight Planarization of Size $n$ [duplicate]
Problem: Maximum Weight Planarization
Given a weighted non-planar graph with $n$ vertices, and $m = \mathcal O\left(n^2\right)$ edges.
Find the subgraph with $n$ nodes (but possibly removing edges to ...
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1answer
184 views
Finding one face in planar graph
Given a planar graph (represented using adjacency lists) we want to find a set of vertices which are around one (random) face. We know that the graph contains at least one triangle.
How do we find ...
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1answer
203 views
A criterion for the planar graph to have unique dual
I get stuck with the following two criteria both about the uniqueness of plane embeddings of a given planar graph. The first one says that a planar graph admits unique plane embedding iff it is a ...
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0answers
64 views
Closed walk in planar graphs that contains $k$ faces
Input: Planar graph $G$ and its embedding in sphere $\Pi$, edges $e, f \in E(G)$ and integer $k$.
Output: A shortest closed walk (one among possibly many, if exists) in $G$ using $e$ and $f$ which ...
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0answers
103 views
Planar TSP: no node insertion?
Since planar TSP with n nodes is NP-hard, we cannot simply find an optimal solution with n-1 nodes and then insert the remaining node at one of the solution's edges to find the optimal solution of the ...
2
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1answer
83 views
Upper bound on the number of triangles in a planar graph
For any $n \geq 4$, I was able to prove that every Apollonian network has $3n - 8$ triangles. An Apollonian network is a planar graph defined by recursively subdividing a triangle by three smaller ...
3
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1answer
125 views
Partitioning planar graphs without minimizing edge cuts
I am looking for an algorithm that, given an undirected, planar graph $G = (V,E)$ with node weights, meets the following conditions:
Creates balanced (within some margin) $k$ partitions of $V$ ...
3
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1answer
399 views
Creating a 2D map of objects given a sparse matrix of pairwise distances
I have a set of points on the two-dimensional plane, but their locations are not given to me. I am given the distance between some pairs of the points. However, I only know these differences for ...
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0answers
82 views
Is there a fast, “partial planarization” algorithm for non-planar graphs?
On "partial planarization" I understand an algorithm, which tries to reach an optimal, or nearly-optimal solution for non-planar graphs. For example, which minimizes the number of the crossing edges (...
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2answers
677 views
Algorithm to generate all planar graphs
Is there an algorithm which provides a sequence of all simple planar graphs, unique by graph isomorphism? For instance: first all planar graphs with 1 node, then all planar graphs with 2 nodes, etc.
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1answer
898 views
Subgraph isomorphism in planar graphs
I'm a computer engineer trying to understand this Eppstein paper for matching subgraphs in planar graphs.
I'm trying to find subgraph matches to map an application graph (the subgraph) to a network-...
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2answers
325 views
Complexity of 4-coloring a map with constraints
The well-known Four color theorem states that every map which is divided into regions, can be colored using 4 colors such that no two adjacent regions have the same color.
In fact, there exists a ...
