<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type="text/xsl" href="../../xslt/mathml.xsl"?><html xmlns="http://www.w3.org/1999/xhtml" xmlns:m="http://www.w3.org/1998/Math/MathML">
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
    <title>Graph Theory: Industrial Drilling</title>
    <link rel="stylesheet" href="../../css/style_mozilla.css" type="text/css" media="screen"/>
  </head>
  <body>
    <div id="logobarleft">
      <a href="http://links.math.rpi.edu/">
        <img src="../../css/titlebar_left.gif" alt="Project Links Logo"/>
      </a>
    </div>
    <div id="logobarright">
      <img src="../../css/titlebar_right.jpg" alt="Artwork"/>
    </div>
    <div id="titlebar">Graph Theory: Industrial Drilling</div>
    <div id="breadcrumbbar">
			Home &gt; Graph Theory: Industrial Drilling
					&gt; Solving the Drilling Problem
					&gt; Independent Sets, Cliques, etc.</div>
    <div id="sidebar">
      <div class="toolmenu1">
        <a href="banana.html" class="toolmenu1">Crib Sheet</a>
      </div>
      <br/>
      <div class="toolmenu2">
        <a href="banana.html" class="toolmenu2">Library</a>
      </div>
      <br/>
      <div class="toolmenu3">
        <a href="banana.html" class="toolmenu3">Help</a>
      </div>
      <br/>
      <div class="toolmenu4">
        <a href="banana.html" class="toolmenu4">Map</a>
      </div>
      <br/>
      <div class="toolmenu5">
        <a href="problemlist.xml" class="toolmenu5">Problem List</a>
      </div>
      <br/>
      <div class="toolmenu5">
        <a href="../index.html" class="toolmenu5">Module Home</a>
      </div>
      <br/>
      <div class="toolmenu6">
        <a href="http://links.math.rpi.edu/" class="toolmenu6">Links Home</a>
      </div>
      <br/>
      <br/>
      <div class="toc_topic">
        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="page4.xml">Objectives</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="">Foundations</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page5.xml">What is a Graph?</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page7.xml">Edges and Vertices</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page10.xml">A First Theorem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page12.xml">Types of Graphs</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page15.xml">Subgraphs and Supergraphs</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page18.xml">Walks, Paths, Cycles</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page21.xml">Trees</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="">Solving the Drilling Problem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page24.xml">Independent Sets, Cliques, etc.</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page26.xml">Vertex Coloring</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page28.xml">Modeling it as a Graph Theory Problem</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page31.xml">First Lower Bound</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page33.xml">The Inside View</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page34.xml">A Good Coloring</a>
      </div>
      <div class="toc_subtopic">
        <a class="toc_link" href="page37.xml">The Best Coloring</a>
      </div>
      <div class="toc_topic">
        <a class="toc_link" href="page38.xml">What's Next?</a>
      </div>
    </div>
    <div id="sectionbar">
      <div id="spacerright" class="spacerright">-</div>
      <div>
        <a href="page24.xml" class="sections">concepts</a>
      </div>
      <div>
        <a href="" class="sectionsgrayed">discover</a>
      </div>
      <div>
        <a href="" class="sectionsgrayed">apply</a>
      </div>
      <div>
        <a href="page25.xml" class="sectionshighlighted">collaborate</a>
      </div>
      <div>
        <a href="" class="sectionsgrayed">practice</a>
      </div>
      <div id="spacerleft" class="spacerleft">-</div>
    </div>
    <div id="content"><h1>Independent Sets, Cliques, etc.</h1>
				
					<p><div class="problem"><a name="problem22"/><b>Problem
				22:
			</b><br/>
						
						<p>
							The red vertices in this graph form an
							independent set of size 4. Is this set maximal?  Find a
							maximum independent set.
						</p>
						<div class="figure" align="center"><a name=""/><img src="../graphics/graph2b.gif"/><br/><b><i>Figure 23</i></b><br/><br/></div>
						
					</div></p>
					<p><div class="problem"><a name="problem23"/><b>Problem
				23:
			</b><br/>
						
							<p>
								Find a clique of size 3 in the graph. Is it maximal?
								Is it maximum?
							</p>
						
					</div></p>
					<p><div class="problem"><a name="problem24"/><b>Problem
				24:
			</b><br/>
						
							<p>
								Find a largest matching in the graph above.
							</p>
						
					</div></p>
				
			</div>
    <div id="nav">
      <a href="page24.xml" class="nav">back</a>
      <a href="page26.xml" class="nav">next</a>
    </div>
    <div class="copyright">
			Copyright 1999
			Rensselaer Polytechnic Institute. All Rights Reserved.
		</div>
  </body>
</html>
