# Questions tagged [planar-graphs]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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. ...
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### 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 $G-S$ ...
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### 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, ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 <...
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### 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 ...
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### 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?
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $V$ ...
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### 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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### 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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### 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. ...