Questions tagged [planar-graphs]

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3
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2answers
118 views

Maximum planar subgraph problem

Given a graph G I want to find the maximum planar subgraph which is a grid graph. (Because the nodes of this subgraph represent points on a grid). Is there any library in python for finding the ...
2
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1answer
69 views

Restore planar graph from vertex degrees

Suppose you are given a list of vertices (with known positions) and their respective degrees, find any set of non-intersecting edges that satisfies the vertex degrees. Or, in other words, connect the ...
6
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2answers
231 views

What edges are not in a Gabriel graph, yet in a Delauney graph?

It is know that the Gabriel graph of a point set $P \subset \mathbb{R}^2$, $\mathcal{GG}(P)$ is a subset of the corresponding Delauney graph $\mathcal{DG}(P)$, i.e. $\mathcal{GG}(P) \subseteq\mathcal{...
0
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1answer
57 views

How to Draw the planar embedding of a graph?

I am very interested to know how to draw the planar embedding of a graph. For this graph: I cannot find the planar embedding because it is a Peterson graph, which is not planar; but for the following ...
1
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1answer
26 views

Are there any established methods for generating random graphs/networks that are both planar and meshlike?

There are well-defined methods for generating random graphs / networks that satisfy certain properties, including small-world graphs, scale-free networks, and totally random non-planar graphs. I am ...
2
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1answer
48 views

Minimum vertices to remove from a graph so that no path exists between two given vertices anymore

Given an undirected graph $G=\{V, E\}$ with its vertices numbered from $1$ to $V$, given two vertices $s$ and $t$ $(1 \leq s \lt t \leq V)$, what is the minimum number of vertices (except $s$ and/or $...
3
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0answers
56 views

Cost of finding optimal elimination order in a planar tensor network?

Suppose we are computing a sum over $n$ factors which can be represented as a planar tensor network. What is the complexity of finding an optimal elimination order? For example, take the following ...
2
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1answer
32 views

Intuitive proof that all planar graphs are disk contact graphs

Planar graphs are graphs which can be drawn on the plane without edges crossing. Disk contact graphs are graphs obtained as follows. Place some disks in the plane without overlaps, allowing touching. ...
1
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1answer
50 views

Similarity measures for (geometric) triangulations

In a project I am working on, we are looking at multiple different (optimal with respect to some cost measure) triangulations of a fixed pointset $S$. I would like to cluster similar triangulations. ...
0
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0answers
61 views

Feedback Vertex Set restricted to planar graphs

In Feedback Vertex Set, we are given an undirected graph $G$ and $k \in \mathbb{N}$, and the objective is to decide whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that $G-S$ ...
1
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1answer
100 views

What's an example of a planar graph with two embeddings whose geometric duals are nonisomorphic?

How to prove that the dual of the dual of a connected planar graph $G$ is isomorphic to $G$? In the post linked above, the user "plop" gives a great response where they claim, in particular, ...
4
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0answers
36 views

Planarity testing given an embedding

I am given a connected graph $G$ with some embedding. I want to find a non-deterministic algorithm running in $O(n)$ time to decide whether $G$ with that embedding is a plane graph (i.e, can be drawn ...
4
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0answers
81 views

If a graph has $15$ vertices, one with degree $8$, $6$ with degree $6$, $8$ with degree $4$, is it a planar graph?

The question is as above. I want to prove that there exists a $K_5$ as a subgraph, so this graph is not a planar graph. But I failed. If you can help me, I will be very appreciative.
2
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2answers
302 views

How to prove that the dual of the dual of a connected planar graph $G$ is isomorphic to $G$?

I find in many references the fact that if $G$ is a connected planar graph, then for any embedding, $G^{**} \cong G$. However, all those references either say that this fact is trivial, or give the ...
4
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0answers
46 views

Decomposing planar Hamiltonian graphs

I have the following the statement and I have to prove whether it is true or not. Given a planar and Hamiltonian graph $\mathbb{G} = (V, E)$, show that we can partition $E = E_1 \cup E_2 = E$ so that ...
3
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2answers
164 views

Finding closest edge to a point in a planar graph

I have a point location problem (in a planar graph) with a twist: rather then finding which region the point is located in, I would like to find the closest segment (edge) to a point, ideally with a <...
0
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0answers
12 views

How to remove filaments from a planar graph?

I have a planar graph and I'm trying to implement this algorithm (https://geometrictools.com/Documentation/MinimalCycleBasis.pdf Chapter 4, page 3). For the filament F0(V4, V3, V2), that has one ...
0
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0answers
25 views

planar 1-in-3 sat described as a planar graph for independent set

Given a planar 1-in-3 sat formula, can someone reduce that formula into a graph that asks the question when ever there is an independent set for it, that's also planar?
1
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1answer
65 views

Max independent set in planar graphs PTAS proof

I've been searching a few hours for a proof to Max independent set in planar graphs beeing in PTAS but I couldn't find anything, I'm searching for one without any reductions and I wonder if anyone ...
1
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1answer
188 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 ...
36
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2answers
2k views

Planar regular languages

In my class a student asked whether all finite automata could be drawn without crossing edges (it seems all my examples did). Of course the answer is negative, the obvious automaton for the language $\...
2
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2answers
690 views

Clique-problem for planar graph

I have to show, that the clique problem in planar graphs is in P. I found the answer here here. However I don't get the conclusion This follows already from Kuratowski's theorem: a clique is at ...
5
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1answer
199 views

Near Triangulation Planar Graph

This is the problem I am dealing with: Given a set P of n points in general position, let a graph G be defined as follows: The vertex set is P. Two vertices, a and b, are joined by an edge provided ...
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0answers
94 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 (...
0
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0answers
33 views

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 ...
2
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0answers
69 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 ...
3
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1answer
165 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* ...
4
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0answers
49 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 ...
6
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1answer
767 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 ...
-1
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1answer
90 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 ...
3
votes
1answer
130 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 ...
3
votes
1answer
146 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 ...
2
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0answers
66 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 ...
0
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1answer
56 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 (...
3
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0answers
136 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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0answers
174 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, $K_{3, 3}$'...
2
votes
1answer
284 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 ...
1
vote
1answer
165 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 ...
1
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0answers
148 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_{...
1
vote
1answer
78 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. ...
1
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1answer
97 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?
2
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1answer
59 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 ...
0
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0answers
26 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?
2
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0answers
306 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 ...
3
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1answer
141 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 $V$ ...
6
votes
1answer
135 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 ...
10
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1answer
2k 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 ...
4
votes
1answer
1k 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. ...
0
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0answers
50 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://...
2
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1answer
231 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 ...