{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ebc44493-a736-44e5-bb72-0a58336c7075",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f9c8092-cdfc-4c98-919a-e1c1b2e804f4",
   "metadata": {},
   "source": [
    "# Planare Graphen"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05df8030-0b10-48a0-8fc6-71f4437d2ba7",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd86dd02-b2ce-49b0-964c-12bc7aaa5e42",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "<b>Ziel des Notebooks:</b>\n",
    "\n",
    "- Verstehen, was planare Graphen sind\n",
    "- Verfahren kennenlernen, um zu bestimmen, ob ein Graph planar ist\n",
    "- Grundlegende Beweismethoden der Graphentheorie nachvollziehen\n",
    "\n",
    "<br>\n",
    "    \n",
    "<b>Mathematische Tools:<b>\n",
    "- Graphentheorie\n",
    "- Kombinatorik\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c90de5e8-b21b-4305-9da4-87491c142040",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd804632-2c1d-4bfb-866f-2356ee06ad33",
   "metadata": {},
   "source": [
    "Oft trifft man in der Mathematik auf einfache Zusammenhänge, die dann formalisiert werden und zwar simpel in der Idee sind, aber sich als sehr komplex oder teils unentscheidbar - etwa viele in der Informatik angesiedelte Probleme - erweisen. Ersteres trifft auf planare Graphen zu, die Idee dieser ist sehr simpel und intuitiv. Allerdings konnten mit Hilfe verschiedener mathematischer Sätze über die Charaktiersierung planarer Graphen gute Kriterien gefunden werden, um zu entscheiden ob ein Graph planar ist oder nicht, sodass mittlerweile verschiedene Algorithmen zur Lösung dieses Problems existieren."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77c51077-5f8b-49b4-8723-909052f0736d",
   "metadata": {},
   "source": [
    "Weiter kann man sich nun natürlich fragen, wieso eine solche Eigenschaft wie Planarität überhaupt relevant ist. Tatsächlich hat sich gezeigt, dass viele Probleme, die für die Klasse der allgemeinen (ungerichteten) Graphen noch nicht gelöst werden konnten, für planare Graphen aufgrund verschiedener Eigenschaften einfacher zu lösen sind. Für dieses Argument ist zu verstehen, dass viele in der Praxis anzutreffende Graphen planar sind. So etwa im Fall von Schienennetzen, wo Stationen die Knoten und Schienen die Kanten darstellen könnten. Oder etwa bei elektronischen Kleinteilen, wie Leiterplatten, bei denen Schaltkreise auf die Platten gedruckt werden und die elektrischen Verbindungen sich ggf. nicht kreuzen dürfen."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e59fcf4-bb21-48fd-87d1-04724f86b4e1",
   "metadata": {},
   "source": [
    "In diesem Notebook wollen wir uns aber vor allem um die mathematisch/theoretischen Hintergründe von planaren Graphen und besagten Sätzen beschäftigen. Dafür werden wir zunächst die Grundbegriffe von Planarität klären, gemeinsam eines der stärksten Kriterium für Planarität, den Satz von Euler, beweisen und zuletzt Folgerungen von diesem in Form von weiteren Kriterien für Planarität betrachten."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee0376e6-dab4-4172-82fd-d0becc71eff8",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d18ccaa5-d1ac-4908-b4a9-0f82f2a32765",
   "metadata": {},
   "source": [
    "Im Folgenden werden wir uns nur auf einfache (ungerichtet und ohne Mehrfachkanten oder Selbstkanten) Graphen beziehen und ohne Einschränkung (siehe Rückblickaufgabe) zusammenhängende Graphen betrachten."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce4a8490-af18-4069-b436-bb49b9314f3e",
   "metadata": {},
   "source": [
    "Zunächst wollen wir uns mit dem Begriff der Planarität bekannt machen:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f96edf8-3f59-4858-acef-ddf5b1324c98",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definition planarer Graph</b> <br>\n",
    "    Ein Graph $G = (V,E)$ heißt <b>planar</b>, wenn er eine Einbettung in die Ebene besitzt. Das heißt, dass er in die Ebene gezeichnet werden kann, sodass seine Kanten durch Jordan-Kurven repräsentiert werden, welche sich nur in gemeinsamen Eckpunkten schneiden. <br>\n",
    "    Wir nennen einen Graphen, der in die Ebene eingebettet ist, auch einen <b>ebenen</b> Graphen.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "587910ef-1d41-4442-b24b-06bc9708d59d",
   "metadata": {},
   "source": [
    "In obiger Beschreibung kann Jordan-Kurve einfach als Kurve, also Verbindung von zwei Eckpunkten verstanden werden. Also sagt die Definition, dass ein Graph planar ist, wenn er (in die Ebene) gezeichnet werden kann, ohne das sich irgendwelche Kanten überschneiden."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9489bc29-92fb-4524-b0f9-a1d9672330f3",
   "metadata": {},
   "source": [
    "Hier ein einfaches Beispiel eines offensichtlich planaren Graphens:"
   ]
  },
  {
   "attachments": {
    "00dba755-41c9-44c9-b530-2f72d2e9c40b.png": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "9fa496b1-0cf3-429c-8111-919c1027f0da",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:00dba755-41c9-44c9-b530-2f72d2e9c40b.png\" width=\"350\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a94af4a4-7448-4cfc-bff6-6cccafb1fb5a",
   "metadata": {},
   "source": [
    "Natürlich stellt sich bei obiger Definition direkt die Frage, wieso nur eine Einbettung in die Ebene gefordert wird. Anders gefragt, kann es überhaupt sein, dass man einen Graph zugleich so aufzeichnen kann, dass sich die Kanten überschneiden und den gleichen Graphen anders aufzeichnen kann, sodass sich keine Kanten überschneiden?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42ca1b9b-73ce-4c78-ac53-837d65606c99",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Beweise oder widerlege, dass wenn ein Graph nach obiger Definition planar ist, der Graph in die Ebene gezeichnet werden kann, sodass sich Kanten (an nicht Eckpunkten) schneiden.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3dcbcd42-9d3f-4021-989b-eac7adb7b681",
   "metadata": {},
   "source": [
    "Um zu verdeutlichen, was für eine große Rolle Planarität in der Graphentheorie spielt, wollen wir, bevor wir uns der eulerschen Polyederformel widmen, betrachten, wieviele planare Graphen es gibt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d65f10ae-5c8f-43e3-916a-852614fb395e",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Bestimme alle einfachen, zusammenhängenden planaren Graphen mit 1,2,3 und 4 Ecken.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71e08067-9392-4ac1-a3e1-0213ff5c5b04",
   "metadata": {},
   "source": [
    "Tatsächlich steigt die Anzahl der einfachen planaren Graphen [schneller](https://de.wikipedia.org/wiki/Planarer_Graph#Kombinatorik) als exponentiell mit der Anzahl an Knoten."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd04d055-a0b2-406c-8e72-8d8fc940d309",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a68ece8b-1701-4d24-a163-1c9a61cd2185",
   "metadata": {},
   "source": [
    "Um noch besser über planare Graphen sprechen zu können, möchten wir noch ein paar Begriffe einführen, die uns ein paar Argumente vereinfachen werden."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ffd6802-4a80-4f9e-91f2-95da48c21d17",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definitionen Land, Gebiet, Landkarte, Rand, benachbart </b> <br>\n",
    "    Für einen ebenen Graphen G legen wir Folgendes fest: <br>\n",
    "    $\\hspace{1cm}$ 1. Mit G wird die Ebene durch die Kurven in endlich viele zusammenhängende Gebiete zerlegt, die wir <b>Länder </b> von G nennen wollen.<br>\n",
    "    $\\hspace{1cm}$ 2. Bei dieser Zerlegung gibt es genau ein Gebiet - das <b> äußere Gebiet </b> -, das nicht beschränkt ist. <br>\n",
    "    $\\hspace{1cm}$ 3. Ein ebener Graph zusammen mit seinen Ländern heißt <b> Landkarte. </b> <br>\n",
    "    $\\hspace{1cm}$ 4. Die <b>Anzahl der Länder</b> von G bezeichnen wir mit $l(G).$ <br>\n",
    "    $\\hspace{1cm}$ 5. Der <b>Rand</b> eines Gebietes ist die Menge der Kanten, die im Abschluss eines Gebietes liegen. <br>\n",
    "    $\\hspace{1cm}$ 6. Zwei Länder heißen <b> benachbart</b>, wenn sie eine gemeinsame Kante haben.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7851af45-5e64-4424-a2bd-b44017c3537b",
   "metadata": {},
   "source": [
    "Wir wollen nun nochmal das Beispiel des obigen planaren Graphen betrachten und einige der soeben definierten Begriffe festigen:"
   ]
  },
  {
   "attachments": {
    "3c3d68a4-342e-4975-80aa-cd74dc7b5e8b.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "13d4a55f-2b38-41a8-b0e7-60a1453caf3b",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:3c3d68a4-342e-4975-80aa-cd74dc7b5e8b.png\" width=\"350\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9a2bca1-829b-4bbb-884a-b14025190cba",
   "metadata": {},
   "source": [
    "Die Gebiete des obigen Graphen sind A, B und C. Das äußere Gebiet ist hierbei C und die Anzahl der Länder entsprechend drei. Der Rand von A ist $\\{\\{1,2\\},\\{2,4\\},\\{4,1\\}\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "912e2edb-6792-459c-985c-eeeedbec1507",
   "metadata": {},
   "source": [
    "Eine interessante Tatsache, die man sich hier vor Augen führen kann ist, dass jede Kante an maximal zwei Ländern anliegt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "889a6b5a-f1c4-4632-a137-9fdd0b6972ce",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbe2400a-3ccd-406e-a89a-8dc5f178fe98",
   "metadata": {},
   "source": [
    "### Eulersche Polyederformel"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ecc11fa4-b508-49c8-9638-76a28f05cfb3",
   "metadata": {},
   "source": [
    "Als erstes wollen wir die Aussage der eulerschen Polyederformel, auch eulerscher Polyedersatz genannt, an sich betrachten:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bd261aa-01cf-4078-a4f0-66ba0ffb9093",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "<b>Eulersche Polyederformel</b> <br>\n",
    "Es sei $G= (V,E)$ ein zusammenhängender ebener Graph. Dann gilt:\n",
    "    $$ |V(G)| - |E(G)| + l(G) = 2.\n",
    "    $$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3668a63-e6af-42be-b42c-6ba7ac78ff94",
   "metadata": {},
   "source": [
    "Zunächst etwas verblüffend, wie einfach der Satz von Euler aussieht. Tatsächlich erweist sich dieser aber ausgesprochen nützlich in der Charakterisierung von planaren Graphen. In obigem Satz steht $l(G)$ wie in der Definition oben für die Anzahl der Länder des Graphen $G.$ $V(G)$ und $E(G)$ sind einfach präzisere Schreibweisen, dafür dass man  die Mengen der Ecken bzw. Kanten vom Graph $G$ und nicht eines anderen Graphen meint. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee7dc42e-89a1-4cc3-94bd-9fdf6b738ff9",
   "metadata": {},
   "source": [
    "Eine Sache die auffällt ist, dass, aus dem Satz folgt, dass jede Einbettung eines planaren Graphen die gleiche Anzahl von Ländern hat. Machen Sie sich gerne nochmal klar, wie dies aus obigem Satz folgt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66288324-bea0-4626-8695-c44fe774894a",
   "metadata": {},
   "source": [
    "Nun wollen wir uns aber dem Beweis der Eulerschen Polyederformel widmen:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f250e654-e2fb-49ba-8c77-f2e70034ba06",
   "metadata": {},
   "source": [
    "Wir beweisen den Satz per Induktion nach $m = |E(G)| \\in \\mathbb{N} \\cup \\{0\\}.$ Dafür ist es sinnvoll noch einmal eine mögliche Definition des Zusammenhangs zu betrachten. Der Induktionsanfang ist hierbei Ihnen überlassen:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7188448b-e375-4d4e-af3f-1cb09f239768",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definition zusammenhängend </b> <br>\n",
    "    Für einen ungerichteten Graphen $G$ heißen zwei Ecken $a,b \\in V(G)$ <b>zusammenhängend </b>wenn ein Weg von $a$ nach $b$ existiert. Dies definiert auf $V(G)$ eine Äquivalenzrelation. Falls die Äquivalenzrelation nur eine Äquivalenzklasse hat, also jede Ecke von jeder anderen Ecke aus erreichbar ist, heißt $G$ <b>zusammenhängend </b>.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17d172ea-31ca-433c-9813-4f85bf350bf9",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Zeigen Sie für einen beliebigen zusammenhängenden ebenen Graph $G$ mit $m=0$ Kanten, dass dieser die Gleichung aus der Eulerschen Polyederformel erfüllt.<br>\n",
    " \n",
    "Hinweis: Was bedeutet es für einen solchen Graphen, dass er zusammenhängend ist?\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7caaa9bf-f48a-469c-9ed4-bda41894bb3b",
   "metadata": {},
   "source": [
    "Bevor wir uns nun dem Induktionsschritt widmen können, müssen wir noch zwei weitere Definitionen bzw. Charakterisierungen bestimmter Graphen besprechen. Die Klasse der Bäume:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69bc3860-443b-41af-87de-75f9f49fc90f",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definition Baum </b> <br>\n",
    "    Ein zusammenhängender Graph ohne Kreise heißt <b>Baum </b>.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a778f517-f6b5-408a-bbfa-2dae65350b1d",
   "metadata": {},
   "source": [
    "Um diese Definition besser zu verstehen betrachten wir zwei Graphen, wobei es sich um einen Baum (links) und einen Graphen, der kein Baum ist (rechts) handelt:"
   ]
  },
  {
   "attachments": {
    "29e7d04a-3a83-4070-8559-3124bad2e148.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "85cee126-c9ae-4af4-ac2b-bdc34a152ac3",
   "metadata": {},
   "source": [
    "![grafik.png](attachment:29e7d04a-3a83-4070-8559-3124bad2e148.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e203fe17-2321-4fa8-b8bc-910b9bd543e4",
   "metadata": {},
   "source": [
    "Wie in den Zeichnungen suggeriert, können Graphen, die Bäume sind, so gezeichnet werden, dass es eine Wurzel gibt von der verschiedene Zweige abgehen von welchen wieder Zweige abgehen, womit auch die Namensgebung zu rechtfertigen ist."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aef69ea0-8d38-4fa7-9339-fad34d193d1c",
   "metadata": {},
   "source": [
    "Aus dieser einfachen Definition lassen sich bereits viele Eigenschaften über Bäume ableiten, wir wollen uns aber auf die Eigenschaft reduzieren, dass für einen Baum mit mindestens zwei Ecken, mindestens zwei Endecken existieren. Also Ecken, die nur einen Nachbarn haben. Diese werden in Bäumen auch Blätter genannt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fd41bdf-824f-43f9-b61b-b16948b8973c",
   "metadata": {},
   "source": [
    "Sicherlich geht nun aus dieser Definition hervor, dass ein Kreisgraph (wie im Folgenden zu sehen) kein Baum ist:"
   ]
  },
  {
   "attachments": {
    "d4347a9b-cdce-4deb-8d0c-202bcdaa82aa.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "1bd32bbe-fd2e-4ad7-8ea5-50ec7edb6087",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:d4347a9b-cdce-4deb-8d0c-202bcdaa82aa.png\" width=\"250\"/>\n",
    "</div>\n",
    "<sub>Der Graph $C_5.$</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ed20e3a-8aa6-434c-85e5-0fefe4416110",
   "metadata": {},
   "source": [
    "Mit obigen Definitionen können wir sagen, dass $l(C_5) = 2.$ Weiter gilt sicherlich, dass wenn wir eine beliebige Kanten entfernen, der resultierende Graph ein Land/Gebiet weniger hat, also $l(C_5') = 1$ gilt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a79fa5e7-f08f-4071-be72-915c91bd8182",
   "metadata": {},
   "source": [
    "Diese beiden Eigenschaften wollen wir uns nun im Induktionsschritt zu Nutze machen, wobei wir eine Fallunterscheidung vornehmen. <br>\n",
    "Sei also $G$ ein beliebiger zusammenhängender Graph mit $m > 0$ und es gelte die zu zeigende Aussage für bel. $k < m,$ $k \\in \\mathbb{Z}_{\\geq 0}.$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b13d40d-dd75-46b7-b099-9f8f4e73ef73",
   "metadata": {},
   "source": [
    "1. Fall: $G$ ist ein Baum"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "794bc7ea-4c1b-424f-a522-d1d4e3263a34",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Zeigen Sie den Induktionsschritt für den 1. Fall. <br>\n",
    "    \n",
    "Hinweis: Entfernen sie eine - bewusst ausgewählte - Ecke und wenden sie die Induktionsvoraussetzung an.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5aad7844-f6e2-4058-a050-dae77c19275f",
   "metadata": {},
   "source": [
    "2. Fall: $G$ ist kein Baum"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7d8044d-cc51-45e1-9b54-c4b5938b4b90",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Zeigen Sie den Induktionsschritt für den 2. Fall. <br>\n",
    "    \n",
    "Hinweis: Was können wir über $G$ sagen, wenn wir wissen, dass $G$ kein Baum ist? Entfernen sie eine - bewusst ausgewählte - Kante und wenden sie die Induktionsvoraussetzung an.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b88b635f-939f-49db-b15d-637ec7ba4051",
   "metadata": {},
   "source": [
    "Sicherlich stellt die eulerscher Polyederformel eine notwendige Bedingung für Graphen dar, um planar zu sein. Anders gesagt können wir einfach die Kanten, Länder und Ecken einer Einbettung zählen (Kanten und Ecken sind natürlich garnicht von der speziellen Einbettung abhängig) und testen, ob die Formel erfüllt ist. <br>\n",
    "Falls die Gleichheit gilt, sagt uns die eulersche Polyederformel allerdings noch nicht, ob der Graph tatsächlich planar ist. Machen Sie sich dies gerne anhand eines Gegenbeispiels klar."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2491ca6-14ab-4b40-a84b-f84350bfc7cd",
   "metadata": {},
   "source": [
    "Es lassen sich allerdings noch weitere - womöglich etwas nützlichere - notwendige Bedingungen, damit ein Graph planar ist, aus der eulerschen Polyederformel ableiten, wie wir im Folgenden sehen wollen."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27b76837-ba5c-4fd9-a976-7f41c642a1b1",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ce84007-46ff-4186-9d02-7d3f80e55aa2",
   "metadata": {},
   "source": [
    "Um den folgenden Teil zu motivieren, wollen wir einen kleinen Schritt zurücktreten und die planarer Graphen etwas heuristischer betrachten. Es ist naheliegend, dass ein Graph mit weniger Kanten eher planar ist, als ein Graph mit sehr vielen Kanten. Wenn man versucht eine Einbettung in die Ebene zu finden, so ist die eigentliche Aufgabe, alle Kanten so zu zeichnen, sodass diese sich nicht kreuzen. <br>\n",
    "Wenn man also einen planaren Graphen hat, ist es naheliegend zu fragen, wie viele Kanten man hinzufügen kann, bis der Graph nicht mehr planar ist. Diese Frage wollen wir mit der Eigenschaft maximal planar erfassen:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f9bad7d-0947-402e-bd6e-fd0ec32b92e9",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definition maximal planar</b> <br>\n",
    "    Ein planarer Graph $G = (V,E)$ heißt <b>maximal planar</b>, falls für alle nicht adjazenten Ecken $u,v \\in V(G)$ der Graph $G+uv,$ also G mit Hinzufügen der Kante zwischen $u$ und $v,$ nicht mehr planar ist. <br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57993929-7b80-4c36-8574-b6f9e435dc2d",
   "metadata": {},
   "source": [
    "Ein maximal planarer Graph ist also maximal in dem Sinne, dass durch Hinzufügen beliebiger Kanten zum Graphen, der Graph nicht planar ist. Tatsächlich zeigt sich, dass sich maximal planare Graphen sehr gut wie folgt charakterisieren lassen:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "120b8175-5ccc-4469-9dba-88dd8aea028f",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Definition Triangulation der Ebene</b> <br>\n",
    "    Eine <b>Triangulation der Ebene</b> (auch Dreiecksgraph) ist ein planarer Graph, in dessen Einbettung jedes Gebiet (auch das äußere) von einem Dreieck berandet ist.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "788d673f-e727-4906-a06f-3a66321749fc",
   "metadata": {},
   "source": [
    "Anders gesagt hat bei einer Triangulation der Ebene, der Rand jedes Gebiets Mächtigkeit 3."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d6defe4-331b-4de5-bdb5-7c4354b8a026",
   "metadata": {},
   "source": [
    "Diese Definition bringt uns nun einen sehr spannenden Zusammenhang:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d118fc3-d38b-4b19-ba80-d2ac44caabf2",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "<b>Satz über Äquivalenzen mit maximal planar</b> <br>\n",
    "Für einen planaren Graphen $G = (V,E)$ (mit mehr als 2 Ecken) sind äquivalent: <br>\n",
    "    $\\hspace{1cm}$ 1. $G$ ist maximal planar <br>\n",
    "    $\\hspace{1cm}$ 2. $G$ ist eine Triangulation <br>\n",
    "    $\\hspace{1cm}$ 3. $|E| = 3 \\cdot |V| - 6$ <br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "238f4e17-eb44-43a8-a8fb-54528f4582b3",
   "metadata": {},
   "source": [
    "Die Beweise der Implikationen $2 \\implies 3$ und $3 \\implies 1$ geben wir aufgrund ihrer Kürze an: <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7ddb6e1-d6d9-471d-9e1d-f99bbcf4c1fd",
   "metadata": {},
   "source": [
    "\"$2 \\implies 3$\": Sei wie oben $l(G)$ als die Anzahl der Länder von G definiert. Da jede Kante an genau zwei Ländern anliegt, erhalten wir $3 \\cdot l(G) = 2 \\cdot |E|,$ also $l(G) = \\frac{2}{3} |E|.$ Mit der eulerschen Polyeder formel folgt nun $2 = |V| - |E| + \\frac{2}{3} |E|.$ Also $|E| = 3 |V| - 6.$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9ae7c0d-c6aa-4cdc-bfe5-e8461619a2cb",
   "metadata": {},
   "source": [
    "\"$3 \\implies 1$\": Angenommen $G$ wäre nicht maximal planar. So kann $G$ durch weitere Kanten zu einem maximal planaren Graphen $G'$ ergänzt werden. Dann gilt sicherlich $|E'| \\geq 3 \\cdot |V'| - 6.$ Da allerdings aus maximaler Planarität $|E'| = 3 \\cdot |V'| -6$ folgt, ist dies ein Widerspruch. Somit ist $G$ bereits maximal planar."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9297b9d2-432d-4273-8677-ab1467159f86",
   "metadata": {},
   "source": [
    "Für die Implikation \"$1 \\implies 2$\" wollen wir folgende hinreichende Überlegung betrachten:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6294d036-a1c3-4db3-81b7-ba929ddf6bce",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Überlegen Sie sich, wieso ein planarer Graph, der ein Gebiet hat, welches von mehr als 3 Kanten berandet wird, nicht maximal planar sein kann.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "456a18f3-be1a-4267-b4b7-0b7ea02acbf1",
   "metadata": {},
   "source": [
    "Mit obigen Äquivalenzen erhalten wir nun:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19a46313-5a4d-4a75-948a-efe09a08a1ab",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b>Notwendiges Kriterium für Planarität eines Graphen</b> <br>\n",
    "    Fur jeden planaren Graphen $G = (V,E)$ gilt $|E| \\leq 3 \\cdot |V| - 6$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9112f6b5-67b3-4a4b-af64-c640b6b08ae4",
   "metadata": {},
   "source": [
    "Als letzte Aufgabe wollen wir nun für einen bekannten Graphen zeigen, dass dieser nicht planar sein kann:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4f66ecf-5431-4652-b0e3-b2cc8c8cdfc7",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "***\n",
    "Zeigen Sie, dass der $K_5,$ der vollständige Graph mit 5 Ecken, nicht planar sein kann.\n",
    "    \n",
    "Hinweis: Unter diesem Block findet sich eine Zeichnung des $K_5$\n",
    "</div>"
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "de2234c7-015f-4e76-a158-6148f878e775",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:bed4e291-ee9c-423e-88d9-9e1d06d0bd7c.png\" width=\"250\"/>\n",
    "</div>\n",
    "<sub>Der Graph $K_5.$</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c812d080-2c32-4916-8395-01c467bc2443",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fe8bacd-ecce-483c-a11e-f6c8aaa891ff",
   "metadata": {},
   "source": [
    "Wie Sie sicherlich schon festgestellt haben, haben wir noch immer kein hinreichendes Kriterium gefunden, um zu beurteilen, ob ein gegebener Graph planar ist. Sicherlich können die Behauptungen vom Anfang über die Lösbarkeit des Planaritätsproblems nicht stimmen, falls ein solches Kriterium nicht existiert."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c563414-63a1-4c5e-ac38-9333e47a2228",
   "metadata": {},
   "source": [
    "Tatsächlich hat bereits im Jahr 1930 Kuratowski, ein polnischer Mathematiker der unter anderem auch durch das Lemma von (Kuratowski-)Zorn bekannt ist, das bis heute bekannteste Kriterium für Planarität geliefert:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1375d195-d155-4fae-9667-a11ec4716bdb",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b>Satz von Kuratowski</b> <br>\n",
    "    Ein schlichter Graph $G$ ist genau dann planar, wenn er keinen Teilgraphen enthält, der eine Unterteilung des $K_5$ oder des $K_{3,3}$ enthält.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b770ab6-6e65-42ac-ae80-081b6f262129",
   "metadata": {},
   "source": [
    "Ein Teilgraph ist hierbei ein Graph $U,$ bei dem beliebige Ecken oder Kanten von $G$ weggelassen wurden. Eine Unterteilung $H$ von $G$ ist wiederum ein Graph, der durch sukzessives Einfügen neuer Knoten entsteht."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b803876-e7f9-4f5e-a31c-0112cd40f02e",
   "metadata": {},
   "source": [
    "Hier einmal zwei Unterteilungen $H_1$ und $H_2$ eines Graphen $G$ und der (vollständige bipartite) Graph $K_{3,3}$:"
   ]
  },
  {
   "attachments": {
    "0e0be197-840d-4714-8b3c-f2e4c4031fec.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "e5718412-660e-4e1a-a3c5-a5ef3842a1c6",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:0e0be197-840d-4714-8b3c-f2e4c4031fec.png\" width=\"700\"/>\n",
    "</div"
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "d86499e2-8733-4043-98c1-3d61d30832bb",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:58721cde-b887-485a-ae37-8a15e562ef43.png\" width=\"200\"/>\n",
    "</div"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99f495b2-48ee-442a-b13a-dcf9d4660ab0",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Bonus Aufgabe\n",
    "***\n",
    "Zeigen Sie, dass der Petersen Graph nicht planar sein kann.\n",
    "    \n",
    "Hinweis: Unter diesem Block findet sich eine Zeichnung des Petersen Graphs.\n",
    "</div>"
   ]
  },
  {
   "attachments": {
    "3b945608-c2bf-453c-b3e6-575dd8f1e9b2.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "4fffa445-91d2-4097-ac7b-7c913c6572f9",
   "metadata": {},
   "source": [
    "<div>\n",
    "<img src=\"attachment:3b945608-c2bf-453c-b3e6-575dd8f1e9b2.png\" width=\"250\"/>\n",
    "</div\n",
    "<sub>Der Petersen Graph.</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee65536a-9e1e-4cd5-8e3f-c36a53cddb0c",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b2dfc9f-3006-45f2-8290-1a1e24704a3a",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "### Rückblick-Aufgabe\n",
    "***\n",
    "Das Ziel dieses Projektes war es zu verstehen, was es bedeutet, wenn ein Graph planar ist und wie dies festgestellt werden kann. Ganz am Anfang wurde gesagt, dass ohne Einschränkung angenommen werden kann, dass die Graphen zusammenhängend sind. Wieso ist dies keine Einschränkung? <br>\n",
    "Weiter noch, wie lässt sich die eulersche Polyederformel auf nicht zusammenhängende Graphen übertragen - recherchieren Sie dies gegebenenfalls gerne.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "094236d8-a83f-4b99-9882-c266e9e0bd14",
   "metadata": {},
   "source": [
    "### Literaturhinweise\n",
    "- Planare Graphen. (2023). In <i>Wikipedia, The free encyclopedia.</i> https://en.wikipedia.org\n",
    "- B. Bollobas: <i>Modern Graph Theory</i>. Springer-Verlag New York (2002)\n",
    "- R. Diestel: <i>Graphentheorie</i>. Springer-Verlag Berlin (2010)\n",
    "- L. Volkmann: <i>Graphen an allen Ecken und Kanten</i> (2006)"
   ]
  }
 ],
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