Data Structures and Algorithms with Object-Oriented Design Patterns in C#
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Undirected Graphs

An undirected graph is a graph in which the nodes are connected by undirected arcs  . An undirected arc is an edge that has no arrow. Both ends of an undirected arc are equivalent--there is no head or tail. Therefore, we represent an edge in an undirected graph as a set rather than an ordered pair:

Definition (Undirected Graph)  An undirected graph   is an ordered pair tex2html_wrap_inline70252 with the following properties:
  1. The first component, tex2html_wrap_inline70254, is a finite, non-empty set. The elements of tex2html_wrap_inline70254 are called the vertices of G.
  2. The second component, tex2html_wrap_inline70260, is a finite set of sets. Each element of tex2html_wrap_inline70260 is a set that is comprised of exactly two (distinct) vertices. The elements of tex2html_wrap_inline70260 are called the edges of G.

For example, consider the undirected graph tex2html_wrap_inline70498 comprised of four vertices and four edges:

eqnarray48551

The graph tex2html_wrap_inline70500 can be represented graphically as shown in Figure gif. The vertices are represented by appropriately labeled circles, and the edges are represented by lines that connect associated vertices.

   figure48555
Figure: An undirected graph.

Notice that because an edge in an undirected graph is a set, tex2html_wrap_inline70504, and since tex2html_wrap_inline70506 is also a set, it cannot contain more than one instance of a given edge. Another consequence of Definition gif is that there cannot be an edge from a node to itself in an undirected graph because an edge is a set of size two and a set cannot contain duplicates.

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Bruno Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.