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Chapter 15: Problem 3

Determine whether the following graphs are planar or not: A complete graph made of four nodes \- A complete graph made of five nodes \- A bipartite graph made of a two-node group and a three-node group in which all possible inter-group edges are present \- A bipartite graph made of two three-node groups in which all possible intergroup edges are present

Short Answer

Expert verified
The graphs \(K_4\) and the two-three bipartite graph are planar. \(K_5\) and \(K_{3,3}\) are not planar.

Step by step solution

01

Understand Graph Planarity

A graph is planar if it can be drawn on a plane without any edges crossing. The key criteria for planarity include Euler's formula for connected planar graphs: \(V - E + F = 2\), where \(V\) is the number of vertices, \(E\) is the number of edges, and \(F\) is the number of faces. Additionally, it must not contain a subgraph homeomorphic to \(K_5\) (the complete graph on 5 vertices) or \(K_{3,3}\) (the complete bipartite graph on 6 vertices, 3 in each group).
02

Analyze Complete Graph with Four Nodes \((K_4)\)

The complete graph \(K_4\) has 4 vertices and \(\binom{4}{2} = 6\) edges. By Euler's formula, \(4 - 6 + 4 = 2\), which satisfies planarity. There is also no subgraph that is homeomorphic to \(K_5\) or \(K_{3,3}\). Thus, \(K_4\) is planar.
03

Analyze Complete Graph with Five Nodes \((K_5)\)

The complete graph \(K_5\) has 5 vertices and \(\binom{5}{2} = 10\) edges. Plugging into Euler's formula, \(5 - 10 + F = 2\) leads to \(F = 7\). However, \(K_5\) cannot be drawn without crossings on a plane and is known not to be planar, as it contains itself and is not homeomorphic to any planar graph.
04

Analyze Bipartite Graph with Two and Three Nodes

In a bipartite graph with groups of two nodes and three nodes, all possible inter-group edges lead to \(2 \times 3 = 6\) edges. Euler鈥檚 formula shows \(5 - 6 + F = 2\) or \(F = 3\). Since this graph does not contain a subgraph homeomorphic to \(K_5\) or \(K_{3,3}\), it is planar.
05

Analyze Bipartite Graph with Two Groups of Three Nodes Each

This graph, denoted as \(K_{3,3}\), has 6 vertices and \(3 \times 3 = 9\) edges. Applying Euler鈥檚 formula gives \(6 - 9 + F = 2\), hence \(F = 5\). However, \(K_{3,3}\) is well-known for not being planar and is used as a typical non-planar example because its structure cannot fit on a plane without edge intersections.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Planar Graphs
A planar graph is a special type of graph that can be drawn on a flat surface where no two edges overlap or intersect unless at a vertex. This is an important concept because it helps us understand how complex connections can be simplified in two-dimensional space. Examples include maps and circuit designs. If a graph can be arranged on a plane without any edge crossings, it is considered planar. Not all graphs have this property. To determine if a graph is planar, we use Euler's formula and check for certain configurations that signal non-planarity, like subgraphs homeomorphic to the complete graph with five nodes or the complete bipartite graph with 3 nodes in each set.
Euler's Formula
Euler鈥檚 formula is central to studying planar graphs. It states that for any connected planar graph, the relationship between vertices (V), edges (E), and faces (F) is given by: \[ V - E + F = 2 \] In this formula, faces include the infinite region that surrounds the graph on the plane. Euler's formula helps confirm if a drawing of a graph in the plane is correct and aids in simplifying the checking of planarity by confirming whether the said relationship holds. If it doesn't, the graph is not planar.
Homeomorphism
Homeomorphism is a concept used to determine when two graphs are essentially the same for planarity purposes. Two graphs are homeomorphic if they can be transformed into one another by adding or removing certain vertices and edges but without breaking their fundamental structure. Critical homeomorphisms relevant to planarity are transformations involving the complete graph on five nodes (K_5) and the complete bipartite graph on 3 nodes each (K_{3,3} ). If a graph contains any of these or their homeomorphic equivalents as subgraphs, it is not planar. This property is frequently used to identify non-planar graphs.
Complete Graph
A complete graph, symbolized as K_n, describes a simple graph where every pair of distinct vertices is connected by a unique edge. In practical terms, each node connects to every other node. The number of edges in a complete graph with n vertices is calculated using the combination formula: \[ \binom{n}{2} = \frac{n(n-1)}{2} \] Examples include K_4, which is planar, and K_5, which isn't. The planarity of complete graphs depends on the number of vertices, with five being the threshold beyond which a complete graph no longer can be planar.
Bipartite Graph
A bipartite graph is a graph whose vertices can be divided into two disjoint sets. No edges exist within each set; they only connect vertices from one set to another. This kind of graph is often used to solve problems involving pairs or matchings. A complete bipartite graph is denoted as K_{m,n} , where each vertex in the first set (m vertices) connects to every vertex in the second set ( n vertices). An example is K_{3,3} , which consists of two sets of three vertices and is known as a classic case of a non-planar graph due to its inability to be represented on a plane without intersecting edges unless allowed to pass over or under each other.

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Most popular questions from this chapter

Write the network data of Zachary's Karate Club graph into a file in each of the following formats: \- Adjacency list \- Edge list

Represent a small social network around you (say, 10 people) as either Graph or DiGraph object of NetworkX. Then add properties to nodes and edges, such as: Node properties: full name, age, job, address, etc. Edge properties: relationship, connection weight, etc.

Create Zachary's Karate Club graph using NetworkX's built-in function, and inspect its nodes and edges to see if they have any non-topological properties.

Every connected tree graph made of \(n\) nodes has exactly \(n-1\) edges. Explain why.

Discuss which type of graph should be used to model each of the following networks. Make sure to consider (a) edge directedness, (b) presence of edge weights, (c) possibility of multiple edges/self-loops, (d) possibility of connections among three or more nodes, and so on. \- Family trees/genealogies \- Food webs among species \- Roads and intersections \- Web pages and links among them \- Friendships among people \- Marriages/sexual relationships \- College applications submitted by high school students \- Email exchanges among coworkers \- Social media (Facebook, Twitter, Instagram, etc.)
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