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Chapter 8: Problem 20

Show that the dual of a planar map is a planar graph.

Short Answer

Expert verified
As the original graph is planar, which means it can be drawn without edges crossing. In its dual, each vertex represents a face in the original graph and each edge represents edges that faces share in the original graph. When we construct the dual graph, we do not introduce any new crossing of edges, which make the dual graph is also a planar graph.

Step by step solution

01

Understand What is a Dual Graph

The dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a loop for each edge that is a loop. Therefore, the faces of G correspond to the vertices of its dual, and the edges of G correspond to the edges of its dual.
02

Planar Map and Its Properties

A planar map is a graph embedded in the plane such that no edges intersect except at their endpoints. In other words, it can be drawn on a piece of paper without any of its edges crossing. Every planar map divides the plane into regions called faces.
03

Proving the Dual of a Planar Map is Planar

From the definition of a dual graph, we can see that an edge in the dual graph corresponds to a face in the original graph and vice versa. Given the planar map is drawn without edges crossing, we can construct its dual by putting a vertex in each face, and connecting these vertices if their corresponding faces share an edge in the original graph. Thus, when we draw the dual, we do not introduce any crossing edges, making the dual also a planar graph.

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Key Concepts

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

Dual Graph
A dual graph is a fascinating concept in graph theory. It can be understood as a transformation of a planar graph. For each face or region of the original planar graph, a vertex is created in the dual graph. The edges are drawn between these vertices such that if two faces of the original graph were adjacent (sharing an edge), there will be a corresponding edge between the vertices in the dual graph.

Key points to remember about dual graphs include:
  • Every vertex in the dual graph corresponds to a face in the original graph.
  • Edges are drawn between vertices in the dual graph for each edge the faces share in the original graph.
  • The dual graph can contain loops and multiple edges if there are faces sharing more than one edge or self-intersecting faces in the original graph.
This relationship beautifully captures the spatial connectivity of regions and boundaries within the original graph.
Planar Graph
A planar graph is a special type of graph that can be explained through its drawing on a flat surface. It can be laid out in such a way that its edges only intersect at their endpoints. This quality makes them essential for many practical applications, such as circuit design and geospatial mapping.

Characteristics of planar graphs include:
  • Can be drawn on a plane surface without edge crossings.
  • Divide the plane into regions called faces.
  • Meet Euler's formula, which states that for a connected planar graph with \(V\) vertices, \(E\) edges, and \(F\) faces, the equation \(V - E + F = 2\) holds.
Planar graphs form the basis for understanding maps and dual graphs, making them a cornerstone in graph theory.
Planar Map
A planar map is more than just a graph; it's an intricate depiction of connectivity on a plane. It showcases how different regions (or faces) are divided and connected through boundaries, represented by edges.

Planar maps are beautiful for several reasons:
  • They provide a real-world model of geographical and abstract spaces.
  • All edges are non-intersecting save for their connecting vertices.
  • Each enclosed region on the map is considered a face, adding spatial dimension to the graph.
This illustrates the idea of graph embeddings, where the map serves as a feasible representation displaying non-crossing edges.
Graph Embedding
The process of graph embedding involves drawing a graph on a surface, such as a plane, in a manner that respects certain constraints, usually no edge crossovers except at vertices.

Simple embedding rules include:
  • Vertices are placed within specific regions (or faces).
  • Edges are drawn such that only intersect at their endpoints.
  • The intent is to maintain the graph's structural integrity without unnecessarily complicating intersections.
Graph embedding is crucial for converting abstract graph concepts into visual and tangible objects, helping in situations like visualization in computer networks, board games, or circuit designs.

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Most popular questions from this chapter

Draw a graph having the given properties or explain why no such graph exists. Four vertices having degrees 1,2,3,4

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When does the complete graph \(K_{n}\) contain an Euler cycle?
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