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<html>
<head>
<title>CaGe -- the quadrangulations windows</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<link href="style.css" rel="stylesheet" type="text/css">
<script language="JavaScript" src="select.js">
</script>
<script language="JavaScript" src="navigate.js">
</script>
<script language="JavaScript" src="subwindow.js">
</script>
</head>
<body onUnload="closeSubwindow ();">
<table border="0" cellspacing="0" cellpadding="0" width="500">
<tr valign="top">
<td>
<h1>The quadrangulations windows</h1>
</td>
<td rowspan="6"><img height="1" width="20" src="Images/trans.gif" alt=""></td>
</tr>
<tr valign="top">
<td>
<p>A quadrangulation is a plane graph with all faces quadrangles.
The Euler formula implies that all quadrangulations must have vertices with degree
at most 3. Furthermore a simple quadrangulations with at least 4 vertices
can not have vertices of degree 1.
</p>
</td>
</tr>
<tr valign="top">
<td>
<h2><br>General Quadrangulations</h2>
<p>
In this window you can generate quadrangulations without too strong restrictions efficiently.
The main invariant that has to be given is the number of vertices. Furthermore the generation
has to be restricted to one of the
following 4 classes: </p>
<ul>
<li>quadrangulations with minimum degree 2 <br>
(for at least 4 vertices these are all simple
quadrangulations)</li>
<li>quadrangulations with minimum degree 3</li>
<li>3-connected quadrangulations</li>
<li>3-connected quadrangulations without 4-cycles that are not the boundary of a face</li>
</ul>
</td>
<td><br>
<a href="Images/cube.png" target="_blank" onClick="return imageWindow (this, 580, 552, 'A cube shown with Rasmol');"><img height="268" width="300" src="Images/cube_small.png" border="0" alt="A cube shown with Rasmol"></a></td>
</tr>
<tr valign="top">
<td>
<h2><br>Quadrangulations with given vertex degrees</h2>
<p> Similar to
<a href="triangulations.html#triachosendeg">triangulations with given vertex degrees</a>,
in this part you can choose the number of vertices and the set of allowed vertex degrees.
Furthermore you can give upper and lower limits for the numbers of vertices of each degree
that must/may be present in the graph.</p>
<p>
The generator used at the moment is designed to work efficiently in case of relatively
many allowed vertex degrees. For the future a reimplementation
of the algorithm described in Discr. Appl. Math. 128, pages 541-554 is planned. This will
be applied in the case of few vertex degrees allowed.
</p>
<td><br>
<a href="Images/quad_given_deg.png" target="_blank" onClick="return imageWindow (this, 568, 530, 'A quadrangulation with given degrees');"><img height="268" width="300" src="Images/quad_given_deg_small.png" border="0" alt="A quadrangulation with given degrees"></a></td>
</tr>
<tr valign="top">
<td align="left"> </td>
</tr>
<tr valign="top">
<td nowrap width="500">
<table border="0" cellspacing="0%" cellpadding=0%" width="100%">
<tr valign="top">
<td align="right" nowrap><a href="generators.html" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);" class="navigation">choosing
a generator window<img src="Images/trans.gif" width=5 height=1 border="0" alt=" "><img src="Images/pointer-backward-passive.gif" width="6" height="11" border="0" alt="<" name="generators"></a></td>
<td nowrap width="100%" align="center"> </td>
<td align="left" nowrap><a href="output.html" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);" class="navigation"><img src="Images/pointer-passive.gif" width="6" height="11" border="0" alt=">" name="output"><img src="Images/trans.gif" width=5 height=1 border="0" alt=" ">output
options</a></td>
</tr>
<tr>
<td colspan="3" bgcolor="#ffffff"><img src="Images/trans.gif" width="1" height="1"></td>
</tr>
<tr valign="top">
<td align="right" nowrap><a href="triangulations.html" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);" class="navigation">the
triangulations windows<img src="Images/trans.gif" width=5 height=1 border="0" alt=" "><img src="Images/pointer-backward-passive.gif" width="6" height="11" border="0" alt="<" name="tubetypes"></a></td>
<td nowrap width="100%" align="center"> </td>
<td align="left" nowrap><a href="polytopes.html" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);" class="navigation"><img src="Images/pointer-passive.gif" id="selecting" width="6" height="11" border="0" alt=">" name="polytopes"><img src="Images/trans.gif" width=5 height=1 border="0" alt=" ">the
general plane graphs windows</a></td>
</tr>
</table>
</td>
</tr>
<tr>
<td><img height="1" width="500" src="Images/trans.gif"></td>
</tr>
</table>
<script language="JavaScript"><!--
document.select_contentName = "generators/quadrangulations";
contentLoaded ();
// --></script>
</body>
</html>