Questions tagged [planar-graphs]

graphs that can be embedded into the plane, i.e. that can be drawn without crossings between the lines representing edges.

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Is there a way to generate all 5-connected 5-regular planar graphs?

My question was partly inspired by the question linked below. There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....? I see a wonderful construction of Adam P. Goucher,...
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5 votes
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386 views

Threshold function for a graph not being planar

A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property. It is well-known that every ...
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Known bounds of the maximum cut of planar graphs

The well-known max cut problem asks for a largest cut of a graph $G$. A cut of maximal size clearly corresponds to a bipartite subgraph of maximal size. After my inquiry, in planar graphs, the maximum-...
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Blind construction of planar graph with additive spanning tree count

Suppose we have two planar graphs $G_1$ and $G_2$ with number of spanning tree count $P_1$ and $P_2$ respectively then there is an easy construction which gives a planar graph with spanning tree count ...
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Completing a tree to a 2-connected outerplanar graph

Let $T$ be a given (finite) tree. Question 1: Is it always possible to add edges to $T$ to obtain a $2$-connected outerplanar supergraph $G$? Question 2: If the answer to Question #1 is negative, can ...
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I don’t understand the two ISOMORPHISM embedding definitions of planar graph in plantri software

The plantri (see http://users.cecs.anu.edu.au/~bdm/plantri/) is a program that generates certain types of graphs that are imbedded on the sphere. Exactly one member of each isomorphism class is output....
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On perfect matchings on planar graphs - is there a linear time deterministic algorithm?

The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree. MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...
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Volume interpretation of number of spanning trees in planar graphs

The number of spanning trees of a planar graph is $DET$ complete where $DET$ class is given in https://complexityzoo.net/Complexity_Zoo:D#det by corollary $21$ in https://mbraverm.princeton.edu/files/...
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Is the divisibility graph of the proper divisors of n more often planar than not?

Define the divisibility graph of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. For all N, is it true that ...
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Does every $4$-connected nonplanar graph contain a $K_5$-minor?

By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K_5$ or $K_{3,3}$. But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not ...
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Chromatic number of rectangle tilings

Suppose we have a region of the plane tiled by finitely many rectangles. We want to color the rectangles so that two rectangles have different colors if they share a part of an edge or if they share ...
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Decomposing a planar graph

Thomassen proved that the vertex set of every planar graph can be decomposed into two sets inducing a 1-degenerate graph and a 2-degenerate graph, respectively (C. Thomassen, Decomposing a planar ...
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An inequality on the number of vertex colorings of planar graphs

Conjecture: Let $G$ be a simple maximal planar graph, and let $P(G,4)$ be the number of proper vertex colorings of $G$ with four colors. Let $v$ be a vertex of $G$ with degree ${\rm deg}(v)=5$, and ...
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Product of vertex degrees of an edge in a planar graph

Let $G$ be a planar graph, which we may assume to be a triangulation, with vertex set $V$ and edge set $E$. Suppose the minimum vertex degree is at least 3, and suppose any two distinct edges share at ...
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The perfect matching problem of planar graph

We know that connectivity is closely related to the Hamiltonian of planar graphs. The most famous result is the Tutte theorem. Theorem (Tutte, 1956). A 4-connected planar graph has a Hamiltonian ...
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Confused about the definition of convex drawing of plane graph

When I looked up the definition of convex drawing of planar graph, my confusion mainly focused on the outer face. The following definition of convex drawing is from Wikipedia. In graph drawing, a ...
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Mac Lane-like condition for intrinsically linked graphs?

If any embedding of your graph in 3-space has two cycles that are linked, then your graph is intrinsically linked (such as the Petersen graph). These graphs generalise non-planar graphs since for ...
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Distribution over Penrose Tilings?

The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow ...
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Can Tutte embedding be guaranteed that each face is convex?

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a ...
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Is graph's planar embedding unique if each block of one planar graph is 3-connected?

A planar graph is one which has a plane embedding. Two drawings are topologically isomorphic if one can be continuously deformed into the other. If we wrap a drawing onto a sphere, and then off again, ...
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Why are trivalent/cubic graphs 'generic' in surfaces?

I've seen some statements that trivalent graphs in a surface are 'generic'. See for example the Wiki entry on cubic graphs. I'm wondering how this could be rephrased. Here are some (somewhat imprecise)...
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Parity of oriented rooted trees

Suppose we have a planar graf with vertices $v_o, \ldots, v_n$, where $n$ is even such that if we checkerboard-color regions in the complement, then the black regions are $n$ (non-degenerated) ...
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Why $K_5$ and $K_{3,3}$?

Most people will have already guessed that this is about Kuratowski's theorem. The theorem states that every non-planar graph must contain a complete graph $K_5$ with five vertices or a complete ...
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is there an example in planar graph that using probabilistic methods

The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the ...
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Equitable 4-colorings of planar triangulations

In an equitable coloring of a graph $G$, the number of vertices in each color class differ by at most $1$. For example, left below is not an equitable coloring, while the right graph is equitably ...
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2 votes
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Two from cubic subgraph hardness

The Problem For a given graph $G$, the cubic subgraph problem asks if there is a subgraph where every vertex has degree 3. The cubic subgraph problem is NP-hard even in bipartite planar graphs with ...
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3 votes
2 answers
218 views

Planar graph of high valence

A classic result in graph theory tells us that any planar graph must have at least one vertex with valence no bigger than 5. On the other hand, there exist examples of planar graphs that are 5-regular ...
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There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....?

Is there any characterization on the set of integers $n$ such that there is a 3-connected 5-regular simple $n$-vertex planar graph?
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What is the standard definition of dual of disconnected planar graph when underlying graph derives 'product structure' over connected graphs?

Dual graph of a plane graph has a standard definition https://en.wikipedia.org/wiki/Dual_graph and an edgeless graph on $n$ vertices is planar. What is the standard dual graph of such a graph? Update ...
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2 votes
1 answer
242 views

Implementation of Koebe–Andreev–Thurston circle packing?

The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the ...
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Planarity of a subgraph

Given a complete symmetric graph $G(V,E)$ with real-valued edgeweights and assume that every induced $k_4$ that is induced by a quadruplet from $v$'s vertices has a unique perfect matching of maximal ...
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12 votes
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Does there exist 2-planar graph with chromatic number 8 or 9 or 10

A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $|E(G)|\le 5(|V(G)|-2)$. This implies that ...
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crossing number and thickness of a simple graph $G$

Crossing number$($cr$)$: The crossing number of a simple graph is the minimum number of crossings that can occur when this graph is drawn in the plane where no three arcs representing edges are ...
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4 votes
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249 views

Reference for results about planar graphs

A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
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Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 14?

Consider a planar graph where every vertex is incident to at least 3 edges, and assign to each edge a weight equal to the sum of the degrees of its endpoints. If not, what is the smallest n so that ...
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The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each ...
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Planar graphs with perfect matching count in linear time?

We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
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2 votes
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117 views

Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
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8 votes
1 answer
340 views

Method to draw 3-connected planar graph on a sphere

The Tutte embedding is a way to create a "nice" drawing of a 3-connected planar graph in the plane, after having chosen an outer face. Is there a similar method to draw such a graph on a sphere? ...
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2 votes
2 answers
135 views

Does any long path in a planar graph contain one of O(n) k-tuple of vertices?

My question is a bit related to both the container method and shallow cell complexity. Let's start with that the number of length $\ell$ paths (where $\ell$ denotes the number of vertices of the path!)...
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4 votes
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451 views

"Locally Nonplanar" graph

A 2-connected $3$-connected graph $G$ is "Almost Planar" Locally Nonplanar if it has a a $2$-connected spanning subgraph $H$ and an embedding in the plane such that $H$ is planar in this embedding and ...
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3 votes
1 answer
323 views

Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...
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4 votes
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94 views

Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
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Reference on generalization of plane graph duality between bonds and simple cycles

Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a ...
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10 votes
1 answer
298 views

Orientations of Planar Graphs

Let $G$ be a $2$-edge-connected graph drawn in the plane (such that the edges intersect only at the endpoints). I want to orient the edges of $G$ such that for each vertex $v$, there are no three ...
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3 votes
0 answers
194 views

Reconstructing plane graphs from degree- and face-sequences

Let $G$ be a plane $3$-connected graph; so it partitions the plane into regions bounded by faces. Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$, and $\mathrm{deg}_f$ be the sequence of ...
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9 votes
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107 views

Partitioning the vertices of a graph into induced trees

I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree. In particular I am ...
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9 votes
2 answers
289 views

Graph planarization via rewiring

Let $G$ be a nonplanar graph (undirected) of $n$ nodes and $e$ edges, with $e \le 3n-6$. Define a rewiring move as replacing edge $(a,b)$ with edge $(a,c)$. The result must be a simple graph (no loops,...
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3 votes
1 answer
108 views

A question regarding the all pair shortest paths in weighted planar graphs

What is the time complexity of the fastest known algorithm for the all-pair shortest paths in planar graphs?
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Smallest planar graph with two non-homoemorphic plane embeddings?

Apologies for asking a question which probably has a well-known answer: What is the smallest (not necessarily simple) planar graph with two non-homeomorphic embeddings into the plane? I am ...
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