# Questions tagged [planar-graphs]

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### Representations of Planar Graphs in Coq

I would like to formalize some simple properties of planar graphs in the Coq proof assistant. 1) How are planar graphs formalized in the Coq proof assistant? Is there a "standard" definition that is ...
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### Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
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### Finding a “lowest” path in a graph

I have an undirected graph $G = (E,V)$, $|V|=n$, where each node $v_i$ has a natural number weight. Think of these weights as heights $h_i$. Given two nodes $s$ and $t$, I'd like to find a lowest ...
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### reference clarification: Whitney's theorem on unique embeddability of 3-connected planar graphs?

This is a question about the correct reference for a result that seems to appear frequently in the literature on planar graph isomorphism. In "A $V \log V$ Algorithm for Isomorphism of Triconnected ...
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### Making planar graph biconnected

I've given a planar graph (with clockwise ordering of vertices). I would like to make it biconnected by adding some number of edges. Graph should remain planar, of course. How can I do this? Few more ...
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### Describe the condition of “non-adjacent 3-cycles” in terms of the cubic adjacency matrix

Oleg Borodin and André Raspaud "A sufficient condition for planar graphs to be 3-colorable" Journal of Combinatorial Theory B88, 2003, 17–27 state the following conjecture: Conjecture 1.2: ...
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### What can i learn about a graph about which only certain properties are known [closed]

1) Suppose we are given the following facts about a graph. What can we conclude/compute beyond these facts? The fact that graph $G(V,E)$ is planar, and thus that it is 4-colorable, The degree of each ...
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### Shortest cycle separator for biconnected planar graphs

An $(s,f)$- balanced separator in a graph $G$ is a set $S$ of $s$ vertices removing which yields connected compoennts of size at most $f|V|$. If the vertices of $S$ form a cycle of length $s$, $S$ is ...
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### Smallest vertex cover which is also an independent set

The vertex cover and independent set as a subset of nodes are always considered in a dual relationship. Have they been looked at together? What I mean is: start from a minimum vertex cover, and if it ...
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### Subclasses or characterizations of modular or pseudo-modular planar graphs

We say that a graph G is modular if for every three vertices x,y,z there exists a vertex w that lies on a shortest path between every two of x, y, z. Pseudo-modular (or "3-Helly") graphs are defined ...
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### Efficient algorithm for testing planarity of the union of two planar graphs

Let $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ both be planar graphs. Is there an efficient algorithm to check whether the union $G = (V,E_1\cup E_2)$ is planar? That is, an algorithm more efficient than ...
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### Graph planarity testing via adjacency matrix

I have looked at several efficient graph planarity algorithms which rely on computing and traversing DFS trees (that add one vertex/edge/path at a time). I am looking for graph planarity algorithms ...
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### Vertices adjacent to Exterior region of a Planar Graph(Algorithm)

Problem: I am looking for an algorithm which finds all vertices that are adjacent to exterior region of a planar graph(For a planar graph, any region=face can be considered as the exterior region /...
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### Shortest non-crossing geometric paths

I have a plane graph $G$ and a set of $k$ vertex pairs $\{s_1,t_1\}, \dots, \{s_k, t_k\}$. The goal is to find $k$ non-crossing paths connecting the pairs of terminals $s_i$ with $t_i$ in the graph so ...
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### How much is known about coloring of planar graphs with degree bounds?

Are there any references that address the following (open?) questions: 1) Is there an algorithm that 4-colors any planar graph of maximal degree at most 5 in linear time? 2) What is the largest ...
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### Sparser Bipartite graphs?

Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members. Let $\mathcal{C}$ be a ...
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### Book Embedding Duality of Graphs

The definition of Book Embedding on Wikipedia: "A book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as ...
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### s-t connectivity on infinite planar graphs with finite description

I would like to know if the following problem is known and has been studied: Consider an infinite directed graph that can be built on the infinite lattice "tiling" a finite set of subgraphs, more ...
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### Time complexity of counting triangles in planar graphs

Counting triangles in general graphs can be done trivially in $O(n^3)$ time and I think that doing much faster is hard (references welcome). What about planar graphs? The following straightforward ...
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### Partition planar graph into connected subgraphs of equal size

Work Jünger, Michael, Gerhard Reinelt, and William R. Pulleyblank. "On partitioning the edges of graphs into connected subgraphs." Journal of graph theory 9.4 (1985): 539-549. states that for 4-...
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### Definition of Planar 3-SAT

What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?
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### Interpolating the Tutte polynomial at the values of two hyperbolas

In a MO question basically I asked when the Tutte polynomial of planar graph can be uniquely determined by the polynomially computable values at the special points and at the two hyperbolas. The ...
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### What is simplest polynomial algorithm for PLANARITY?

There are several algorithms that decide in polynomial time whether a graph can be drawn in the plane or not, even many with a linear running time. However, I could not find a very simple algorithm ...
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### A bijection between ordered lambda terms and rooted planar maps?

Consider the following recurrence in two parameters $n$ and $k$: \begin{aligned} NF(0,k) &= 0 \\ NF(n,k) &= Neu(n,k) + NF(n-1,k+1) \\ Neu(n,k) &= [n=1 \wedge k=1] + \sum_{l=1}^{n-1}\sum_{...
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### Dual Barnette's Conjecture

Is every Eulerian triangulated (planar) graph Hamiltonian? On the other hand we have that: Barnette's Conjecture (Open): Every cubic bipartite (3-connected) planar graph is Hamiltonian. Notice ...
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### Number of $k$-cuts of grid graphs

Given a $n\times m$ grid, let the bottom-left vertex be $s$ and the top-right vertex be $t$. Given $k$ non-consecutive edges on the upper horizontal line of the grid, I want to find an upper bound on ...
A planar graph is a graph which can be embedded in the plane, without having crossing edges. Let $G=(X,E)$ be a $k$-uniform-hypergraph, i.e. an hypergraph such that all its hyperedges have size k. ...