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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION0017105000000000000000">Undirected Graphs</A></H3>
<P>
An undirected graph is a graph in which the nodes are connected by
<em>undirected arcs</em><A NAME=49012> </A><A NAME=49013> </A>.
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:
<P>
<BLOCKQUOTE> <b>Definition (Undirected Graph)</b><A NAME="defngraph"> </A>
An <em>undirected graph</em><A NAME=49018> </A><A NAME=49019> </A>
is an ordered pair <IMG WIDTH=72 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71355" SRC="img2282.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2282.gif" > with the following properties:
<OL><LI>
The first component, <IMG WIDTH=10 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71357" SRC="img2283.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2283.gif" >, is a finite, non-empty set.
The elements of <IMG WIDTH=10 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71357" SRC="img2283.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2283.gif" > are called the <em>vertices</em> of <I>G</I>.<LI>
The second component, <IMG WIDTH=9 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71363" SRC="img2284.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2284.gif" >, is a finite set of sets.
Each element of <IMG WIDTH=9 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71363" SRC="img2284.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2284.gif" > is a set that is comprised of
exactly two (distinct) vertices.
The elements of <IMG WIDTH=9 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline71363" SRC="img2284.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2284.gif" > are called the <em>edges</em> of <I>G</I>.
</OL></BLOCKQUOTE>
<P>
For example, consider the undirected graph <IMG WIDTH=91 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71601" SRC="img2332.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2332.gif" >
comprised of four vertices and four edges:
<P> <IMG WIDTH=500 HEIGHT=40 ALIGN=BOTTOM ALT="eqnarray49025" SRC="img2333.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2333.gif" ><P>
The graph <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline71603" SRC="img2334.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2334.gif" > can be represented
<em>graphically</em> as shown in Figure <A HREF="page529.html#figgraph3" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page529.html#figgraph3"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
The vertices are represented by appropriately labeled circles,
and the edges are represented by lines
that connect associated vertices.
<P>
<P><A NAME="49154"> </A><A NAME="figgraph3"> </A> <IMG WIDTH=575 HEIGHT=136 ALIGN=BOTTOM ALT="figure49029" SRC="img2335.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2335.gif" ><BR>
<STRONG>Figure:</STRONG> An Undirected Graph<BR>
<P>
<P>
Notice that because an edge in an undirected graph is a set,
<IMG WIDTH=94 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline71607" SRC="img2336.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2336.gif" >,
and since <IMG WIDTH=15 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline71609" SRC="img2337.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img2337.gif" > is also a set,
it cannot contain more than one instance of a given edge.
Another consequence of Definition <A HREF="page529.html#defngraph" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page529.html#defngraph"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> 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.
<P>
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