# Questions tagged [planar-graphs]

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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 ...
440 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 ...
911 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-...
693 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. ...
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 ...
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 ...
757 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. ...
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 $V$ ...
51 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* ...
64 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 ...
65 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 ...
407 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 ...
77 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 ...
111 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, $K_{3, 3}$'...
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 ...
99 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 ...
49 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 ...
84 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 ...
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 ...
176 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 ...
24 views

### 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 ...
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 ...
335 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 ...
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 ...
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?
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 ...
206 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 ...
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 ...