tait
/
tait.tech
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity
You can not select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
Branches Tags
${ item.name }
Create tag ${ searchTerm }
Create branch ${ searchTerm }
from 'ca770be68f'
${ noResults }
tait.tech/_site/transcriptions/src/cmpt-225/08/08/index.html

1128 lines
64 KiB
Raw Blame History Escape

This file contains invisible Unicode characters!

This file contains invisible Unicode characters that may be processed differently from what appears below. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to reveal hidden characters.

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title> | tait.tech</title>
<link rel="stylesheet" href="/assets/css/style.css" id="main-css">
<link rel="stylesheet" href="/assets/css/transcription.css" id="trans-css">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script src="/assets/js/"></script>
<link rel="stylesheet" href="/assets/css/katex.css" id="math-css">
</head>
<body>
<main>
<div id="wrapper">
<h1 id="rooted-trees--cmpt-225">Rooted Trees – CMPT 225</h1>
<h2 id="graphs">Graphs</h2>
<p>Graph is a pair G=&lt;V,E&gt;, with:</p>
<ul>
<li>V, a set called “vertacies” or “nodes”</li>
<li>E, a set of pairs from V, i.e. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>⊆</mo><mi>V</mi><mo>×</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">E \subseteq V\times V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span> called edges</li>
<li>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">G=\langle\{1,2,3\},\{(1,2),(1,3)\}\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">G</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></li>
</ul>
<p>Example in table format:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g1-1">1</span></td>
<td><a href="#g1-2">2</a>,<a href="#g1-3">3</a></td>
</tr>
<tr>
<td><span id="g1-2">2</span></td>
<td><a href="#g1-1">1</a></td>
</tr>
<tr>
<td><span id="g1-3">3</span></td>
<td><a href="#g1-1">1</a></td>
</tr>
</tbody>
</table>
<p>“G is directed”: edges are ordered pairs (often called arcs)</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo><mo mathvariant="normal">≠</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">
\langle\{\{1,2,3\},\{(1,2),(1,3)\}\}\rangle
\neq
\langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></span></p>
<p>i.e. graph one:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g2-1">1</span></td>
<td><a href="#g2-2">2</a>,<a href="#g2-3">3</a></td>
</tr>
<tr>
<td><span id="g2-2">2</span></td>
<td> </td>
</tr>
<tr>
<td><span id="g2-3">3</span></td>
<td> </td>
</tr>
</tbody>
</table>
<p>does not equal graph two:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g3-1">1</span></td>
<td><a href="#g3-3">3</a></td>
</tr>
<tr>
<td><span id="g3-2">2</span></td>
<td><a href="#g3-1">1</a></td>
</tr>
<tr>
<td><span id="g3-3">3</span></td>
<td> </td>
</tr>
</tbody>
</table>
<p>“G is undirected” edges are sets</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">
\langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle
=
\langle\{\{1,2,3\},\{(1,2),(3,1)\}\}\rangle
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></span></p>
<p>i.e. graph one:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g4-1">1</span></td>
<td><a href="#g4-3">3</a>,<a href="#g4-2">2</a></td>
</tr>
<tr>
<td><span id="g4-2">2</span></td>
<td><a href="#g4-1">1</a></td>
</tr>
<tr>
<td><span id="g4-3">3</span></td>
<td><a href="#g4-1">1</a></td>
</tr>
</tbody>
</table>
<p>is equal to graph two:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g5-1">1</span></td>
<td><a href="#g5-2">2</a>,<a href="#g5-3">3</a></td>
</tr>
<tr>
<td><span id="g5-2">2</span></td>
<td><a href="#g5-1">1</a></td>
</tr>
<tr>
<td><span id="g5-3">3</span></td>
<td><a href="#g5-1">1</a></td>
</tr>
</tbody>
</table>
<p>By default, by “graph” we will mean “undirected graph”.</p>
<h2 id="path">Path</h2>
<p>Path of G of length n is a sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},V_{1},V_{2},\dots V_{n}\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> of vertacies s-t. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">(V_{0},V_{1}),(V_{1},V_{2}),(V_{2},V_{3})\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span></span></span></span> are edges of G.</p>
<p>s-t Path in G: path <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,\dots,t\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">t</span><span class="mclose">⟩</span></span></span></span> in G.</p>
<p>Example of a s-t path of length 6 for G: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>e</mi><mo separator="true">,</mo><mi>f</mi><mo separator="true">,</mo><mi>h</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,e,f,h,t\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">t</span><span class="mclose">⟩</span></span></span></span></p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g6-a">a</span></td>
<td><a href="#g6-b">b</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-b">b</span></td>
<td><a href="#g6-a">a</a>,<a href="#g6-c">c</a></td>
</tr>
<tr>
<td><span id="g6-c">c</span></td>
<td><a href="#g6-b">b</a>,<a href="#g6-d">d</a>,<a href="#g6-e">e</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-d">d</span></td>
<td><a href="#g6-c">c</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-e">e</span></td>
<td><a href="#g6-c">c</a>,<a href="#g6-f">f</a>,<a href="#g6-g">g</a></td>
</tr>
<tr>
<td><span id="g6-f">f</span></td>
<td><a href="#g6-e">e</a>,<a href="#g6-h">h</a></td>
</tr>
<tr>
<td><span id="g6-g">g</span></td>
<td><a href="#g6-e">e</a>,<a href="#g6-h">h</a>,<a href="#g6-t">t</a></td>
</tr>
<tr>
<td><span id="g6-h">h</span></td>
<td><a href="#g6-f">f</a>,<a href="#g6-g">g</a>,<a href="#g6-t">t</a></td>
</tr>
<tr>
<td><span id="g6-s">s</span></td>
<td><a href="#g6-a">a</a>,<a href="#g6-c">c</a>,<a href="#g6-d">d</a></td>
</tr>
<tr>
<td><span id="g6-t">t</span></td>
<td><a href="#g6-g">g</a>,<a href="#g6-h">h</a></td>
</tr>
</tbody>
</table>
<p>Vertex t is <em>reachable</em> from s in G if ther is an s-t path in G.
G is <em>connected</em> if for every pair <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">u,v\in V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>, u is reachable from v.</p>
<p>Transcriber’s note: this is much easier to show with graphics and in fact very hard to show easily without graphics. I did my best, but it should be noted that this is <em>very</em> obvious with a visual graph.
It becomes two seperate graphs, in essense.
So for convenience, I will be splitting this up into two tables, even though technically it’s one “graph” with two sets of connected nodes.</p>
<p id="connected1">Connected:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g7-a">a</span></td>
<td><a href="#g7-b">b</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-b">b</span></td>
<td><a href="#g7-a">a</a>,<a href="#g7-c">c</a></td>
</tr>
<tr>
<td><span id="g7-c">c</span></td>
<td><a href="#g7-b">b</a>,<a href="#g7-d">d</a>,<a href="#g7-e">e</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-d">d</span></td>
<td><a href="#g7-c">c</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-e">e</span></td>
<td><a href="#g7-c">c</a>,<a href="#g7-f">f</a>,<a href="#g7-g">g</a></td>
</tr>
<tr>
<td><span id="g7-f">f</span></td>
<td><a href="#g7-e">e</a>,<a href="#g7-h">h</a></td>
</tr>
<tr>
<td><span id="g7-g">g</span></td>
<td><a href="#g7-e">e</a>,<a href="#g7-h">h</a>,<a href="#g7-t">t</a></td>
</tr>
<tr>
<td><span id="g7-h">h</span></td>
<td><a href="#g7-f">f</a>,<a href="#g7-g">g</a>,<a href="#g7-t">t</a></td>
</tr>
<tr>
<td><span id="g7-s">s</span></td>
<td><a href="#g7-a">a</a>,<a href="#g7-c">c</a>,<a href="#g7-d">d</a></td>
</tr>
<tr>
<td><span id="g7-t">t</span></td>
<td><a href="#g7-g">g</a>,<a href="#g7-h">h</a></td>
</tr>
</tbody>
</table>
<p>Not connected:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g9-a">a</span></td>
<td><a href="#g9-b">b</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-b">b</span></td>
<td><a href="#g9-a">a</a>,<a href="#g9-c">c</a></td>
</tr>
<tr>
<td><span id="g9-c">c</span></td>
<td><a href="#g9-b">b</a>,<a href="#g9-d">d</a>,<a href="#g9-e">e</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-d">d</span></td>
<td><a href="#g9-c">c</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-e">e</span></td>
<td><a href="#g9-c">c</a></td>
</tr>
<tr>
<td><span id="g9-s">s</span></td>
<td><a href="#g9-a">a</a>,<a href="#g9-c">c</a>,<a href="#g9-d">d</a></td>
</tr>
</tbody>
</table>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g10-f">f</span></td>
<td><a href="#g10-h">h</a></td>
</tr>
<tr>
<td><span id="g10-g">g</span></td>
<td><a href="#g10-h">h</a>,<a href="#g10-t">t</a></td>
</tr>
<tr>
<td><span id="g10-h">h</span></td>
<td><a href="#g10-f">f</a>,<a href="#g10-g">g</a>,<a href="#g10-t">t</a></td>
</tr>
<tr>
<td><span id="g10-t">t</span></td>
<td><a href="#g10-g">g</a>,<a href="#g10-h">h</a></td>
</tr>
</tbody>
</table>
<h2 id="cycles--trees">Cycles &amp; Trees</h2>
<p>Cycle in G: path <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots V_{n-1},V_{n}\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> in G where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mo>=</mo><msub><mi>V</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_{0}=V_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p><em>Simple path</em>: all vertecies are distinct.
<em>Simple Cycle</em>: cycle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots V_{n-1},V_{n}\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots,V_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span> is a simple path.</p>
<p>Simple cylcle of length 5: (see <a href="#connected1">“connected” graph</a> for confirmation): <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,b,a,s\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span></p>
<p>Cycle of length 7 + repeats: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>f</mi><mo separator="true">,</mo><mi>e</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,b,f,e,c,s\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span></p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g11-a">a</span></td>
<td><a href="#g11-b">b</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-b">b</span></td>
<td><a href="#g11-a">a</a>,<a href="#g11-c">c</a>,<a href="#g11-f">f</a></td>
</tr>
<tr>
<td><span id="g11-c">c</span></td>
<td><a href="#g11-b">b</a>,<a href="#g11-d">d</a>,<a href="#g11-e">e</a>,<a href="#g11-f">f</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-d">d</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-e">e</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-f">f</a>,<a href="#g11-g">g</a></td>
</tr>
<tr>
<td><span id="g11-f">f</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-e">e</a>,<a href="#g11-h">h</a>,<a href="#g11-b">b</a></td>
</tr>
<tr>
<td><span id="g11-g">g</span></td>
<td><a href="#g11-e">e</a>,<a href="#g11-h">h</a>,<a href="#g11-t">t</a></td>
</tr>
<tr>
<td><span id="g11-h">h</span></td>
<td><a href="#g11-f">f</a>,<a href="#g11-g">g</a>,<a href="#g11-t">t</a></td>
</tr>
<tr>
<td><span id="g11-s">s</span></td>
<td><a href="#g11-a">a</a>,<a href="#g11-c">c</a>,<a href="#g11-d">d</a></td>
</tr>
<tr>
<td><span id="g11-t">t</span></td>
<td><a href="#g11-g">g</a>,<a href="#g11-h">h</a></td>
</tr>
</tbody>
</table>
<h2 id="tree">Tree</h2>
<p>Tree: a connected, acyclic graph:</p>
<p>Example 1 (not a tree, due to cycle: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,c,d,s\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span>):</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g12-a">a</span></td>
<td><a href="#g12-b">b</a></td>
</tr>
<tr>
<td><span id="g12-b">b</span></td>
<td><a href="#g12-a">a</a>,<a href="#g12-c">c</a></td>
</tr>
<tr>
<td><span id="g12-c">c</span></td>
<td><a href="#g12-b">b</a>,<a href="#g12-d">d</a>,<a href="#g12-e">e</a>,<a href="#g12-s">s</a></td>
</tr>
<tr>
<td><span id="g12-d">d</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-s">s</a></td>
</tr>
<tr>
<td><span id="g12-e">e</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-g">g</a></td>
</tr>
<tr>
<td><span id="g12-f">f</span></td>
<td><a href="#g12-h">h</a></td>
</tr>
<tr>
<td><span id="g12-g">g</span></td>
<td><a href="#g12-e">e</a>,<a href="#g12-h">h</a>,<a href="#g12-t">t</a></td>
</tr>
<tr>
<td><span id="g12-h">h</span></td>
<td><a href="#g12-f">f</a>,<a href="#g12-g">g</a>,<a href="#g12-t">t</a></td>
</tr>
<tr>
<td><span id="g12-s">s</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-d">d</a></td>
</tr>
<tr>
<td><span id="g12-t">t</span></td>
<td><a href="#g12-g">g</a>,<a href="#g12-h">h</a></td>
</tr>
</tbody>
</table>
<p>Example 2 (a tree, no cycle, all connected):</p>
<p>Now using ARIA trees:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
s
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
g
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
i
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Example 3 (not a tree) (transcriber’s note: written as two trees for simplicity’s sake)</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
s
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
e
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
h
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
i
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Fact: every tree with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span> verticies has <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> edges.</p>
<h2 id="rooted-tree">Rooted-tree:</h2>
<p>Rooted tree: tree with a distringuished vertex called the root.</p>
<p>Unrooted tree:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
c
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
i
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
j
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
k
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
l
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Tree T with root:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
a (root)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
c
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
i
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
j
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
k
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
l
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>An alternate drawing of T, with no semantic change.</p>
<p>Notice: in a tree there is a unique path between any two verticies.
So: a unique path from any vertex to the root.
Thus: the root indicates a <em>direction</em> on the edges e.g. towards the root.</p>
<p>A graph of the same tree, T, with arrows pointing towards the root.
There is no reason for having any semantic markup for this.</p>
<p>(sometimes “away from the root”).</p>
<h2 id="rooted-tree-terminology">Rooted Tree Terminology</h2>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
z (root, anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
y
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
x (leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
w (leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
v (anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
u (leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
t (anscestor of a, parent of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
s (sibling of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
r (sibling of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
a (descendant of a, anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
q (child of a, desandant of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
o (decendant of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
n (decendant of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
m (decendant of a, leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
p (child of a, desendant of a, leaf)
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<ul>
<li>The <em>root</em> has no parent.</li>
<li><em>Leaves</em> have no children.</li>
<li>Internal nodes are the non-leaves (sometimes root ??? [can’t read] too)</li>
</ul>
<h2 id="depth--height">Depth &amp; Height</h2>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
4 (depth 0)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1 (depth 1)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
3 (depth 1)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
2 (depth 2)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1 (depth 3)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
0 (depth 4)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 3)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<ul>
<li>Depth of node v = length of path from v to the root.</li>
<li>Height of node v = length of longest path from to a decendant of v (e.g. to a leaf)</li>
<li>Height of Tree T = height of its root = max height of any node in T = max depth of any node in T.</li>
</ul>
<p>A rooted tree is:</p>
<ul>
<li>k-ary if <em>no node has</em> &gt;k children.</li>
<li>binary if <em>no node has</em> &gt;2 children.</li>
<li>ordered if the children of every node are ordered.</li>
</ul>
<p>E.g. A ordered trenary tree:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
no name
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
3
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
3
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Notice: when we draw a tree, or represent it in a data structure, we order it.</p>
<p>In an ordered binary tree, every child of a node v is either the “left child of v” or the “right child of v”.</p>
<p>Transcriber’s note: nodes’ children are transcribes from left to right. So for a binary tree, the first child is the “left child” and the second child is the “right child”.</p>
<h2 id="subtree-rooted-at-v">Subtree rooted at v</h2>
<p>Subtree rooted at v: tree with root v and containing all decendants of v.</p>
<p>In a binary tree:</p>
<ul>
<li>“left subtree of v” means the subtree rooted at the left child of v.</li>
<li>sim. for “right child of v”.</li>
</ul>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
z
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
v (subtree rooted at v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
y (subtree rooted at v, left subtree of v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
v (subtree rooted at v, left subtree of v
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
u (subtree rooted at v, left subtree of v
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
x (subtree rooted at v, right subtree of v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
w (subtree rooted at v, right subtree of v)
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<h2 id="end">End</h2>
</div>
</main>
<hr>
<footer>
</footer>
</body>
</html>
Reference in new issue View Git Blame Copy Permalink