GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Algorithmus von Hierholzer</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algorithmus von Hierholzer</span></h1>

</header>

</div>

<p>Der <b>Algorithmus von Hierholzer</b> ist ein <a href="Algorithmus" title="Algorithmus">Algorithmus</a> aus dem Gebiet der <a href="Graphentheorie" title="Graphentheorie">Graphentheorie</a>, mit dem man in einem <a href="Graph_(Graphentheorie)" title="Graph (Graphentheorie)">ungerichteten Graphen</a> einen <a href="Eulerkreisproblem" title="Eulerkreisproblem">Eulerkreis</a> bestimmt. Er geht auf Ideen von <a href="Carl_Hierholzer" title="Carl Hierholzer">Carl Hierholzer</a> zurück.

</p><p>Voraussetzung: Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=(V,E)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/644a8d85ee410b6159ca2bdb5dcb9097e2c8f182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle G=(V,E)}" loading="lazy"></span> ein <a href="Zusammenh%C3%A4ngender_Graph" class="mw-redirect" title="Zusammenhängender Graph">zusammenhängender Graph</a>, der nur <a href="Knoten_(Graphentheorie)" title="Knoten (Graphentheorie)">Knoten</a> mit geradem <a href="Grad_(Graphentheorie)" title="Grad (Graphentheorie)">Grad</a> aufweist.

</p>

<ol><li>Wähle einen beliebigen Knoten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}\in V}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{0}\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e13fb4af38f15f65bd01812c73d8e06b2daca524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle v_{0}\in V}" loading="lazy"></span> des Graphen und konstruiere von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60faad24775635f4722ccc438093dbbfe05f34ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}" loading="lazy"></span> ausgehend einen <a href="Kreisgraph" title="Kreisgraph">Unterkreis</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, der keine Kante in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> zweimal durchläuft.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> ein Eulerkreis von G ist, also alle Kanten von G enthält, breche ab. Andernfalls:</li>

<li>Vernachlässige nun alle <a href="Kante_(Graphentheorie)" title="Kante (Graphentheorie)">Kanten</a> des Unterkreises <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>.</li>

<li>Am ersten Eckpunkt von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>, dessen Grad größer 0 ist, lässt man nun einen weiteren Unterkreis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63399d2b53fc472cfd349fd6167c21585d0a37f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}" loading="lazy"></span> in G entstehen, der keine Kante in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> – also keine schon besuchte Kante – durchläuft und keine Kante in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> zweimal enthält.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> den zweiten Kreis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63399d2b53fc472cfd349fd6167c21585d0a37f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}" loading="lazy"></span> ein, indem in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> der Startknoten von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63399d2b53fc472cfd349fd6167c21585d0a37f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}" loading="lazy"></span> durch alle Knoten von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63399d2b53fc472cfd349fd6167c21585d0a37f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}" loading="lazy"></span> in der richtigen Reihenfolge ersetzt wird.</li>

<li>Nenne jetzt den so erhaltenen Kreis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> und fahre bei Schritt 2 fort.</li></ol>

<p>Die <a href="Komplexit%C3%A4t_(Informatik)" title="Komplexität (Informatik)">Komplexität des Algorithmus</a> ist linear in der Anzahl der Kanten.

</p>

<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>

<p>Gegeben sei ein Eulergraph mit neun Knoten (siehe erste Abbildung). Ein Zyklus vom Startknoten 1 wäre beispielsweise die in der zweiten Abbildung blau dargestellte Knotenfolge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{blau}}=(1,2,3,7,1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{blau}}=(1,2,3,7,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/668c92a2a4a6edebe84d2294399a43b999dfda4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.857ex; height:2.843ex;" alt="{\displaystyle C_{\text{blau}}=(1,2,3,7,1)}" loading="lazy"></span>. Nach Entfernen dieser Kanten haben die Knoten 1, 3 und 7 der bisher gebildeten Zykel noch einen Knotengrad größer Null, welche als Startknoten für den nächsten Zyklus in Frage kommen. Vom Startknoten 3 aus kann man den Kreis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{rot}}=(3,1,8,7,4,3)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{rot}}=(3,1,8,7,4,3)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed7c336dd23b49b205eee6dd9ca5fdda56594e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.053ex; height:2.843ex;" alt="{\displaystyle C_{\text{rot}}=(3,1,8,7,4,3)}" loading="lazy"></span> bilden (in der dritten Abbildung rot). Wählt man nun als Startknoten den Knoten 7, kann man von den übrig gebliebenen Kanten den Zykel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{grün}}=(7,6,9,5,4,9,7)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{grün}}=(7,6,9,5,4,9,7)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56cfb155108977552f8f311ee0758c07b0d3e0ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:24.525ex; height:3.509ex;" alt="{\displaystyle C_{\text{grün}}=(7,6,9,5,4,9,7)}" loading="lazy"></span> bilden. Setzt man jetzt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{grün}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{grün}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/369b4632b64d9c81307eb3cdc27799518ce4bdb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.276ex; height:3.343ex;" alt="{\displaystyle C_{\text{grün}}}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{rot}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{rot}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a50b538d6e2da4fef1d5f2636b794d3e3072879c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4ex; height:2.509ex;" alt="{\displaystyle C_{\text{rot}}}" loading="lazy"></span> an Stelle des Knoten 7 ein, erhält man den Zykel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,1,8,7,6,9,5,4,9,7,4,3)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (3,1,8,7,6,9,5,4,9,7,4,3)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b2edd531f1a1e46dc7fa6a507448e34c11ee44d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.132ex; height:2.843ex;" alt="{\displaystyle (3,1,8,7,6,9,5,4,9,7,4,3)}" loading="lazy"></span>. Setzt man diesen in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{blau}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{blau}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/541777d464f54e7e3a5d9a49a0ef24801c5e7526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.001ex; height:2.509ex;" alt="{\displaystyle C_{\text{blau}}}" loading="lazy"></span> an Stelle des Knoten 3 ein, erhält man die mögliche Eulertour <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,3,1,8,7,6,9,5,4,9,7,4,3,7,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,2,3,1,8,7,6,9,5,4,9,7,4,3,7,1)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e9fdd29187eb9faaa9af6d9c7e1924622e3f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.918ex; height:2.843ex;" alt="{\displaystyle (1,2,3,1,8,7,6,9,5,4,9,7,4,3,7,1)}" loading="lazy"></span> wie in der letzten Abbildung gezeigt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Carl Hierholzer: <i>Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren</i>. Mathematische Annalen VI (1873), 30–32. <a rel="nofollow" class="external autonumber" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002242176">[1]</a></li>

<li>Sven Oliver Krumke, Hartmut Noltemeier: <i>Graphentheoretische Konzepte und Algorithmen.</i> Teubner, Wiesbaden 2005, ISBN 3-519-00526-3, S. 45–48</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://algorithms.discrete.ma.tum.de/graph-algorithms/hierholzer/index_de.html">Applet zur Visualisierung</a></li>

<li><a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/mathematik/eulerscher-graph/2843"><i>Eulerscher Graph.</i></a> In: <i>Springer Lexikon der Mathematik.</i> (mit einer Darstellung des nach Hierholzer benannten Algorithmus)</li>

<li><a href="Oliver_Deiser" title="Oliver Deiser">Oliver Deiser</a>: <a rel="nofollow" class="external text" href="https://www.aleph1.info/?call=Puc&permalink=ema21_2_4_Z2"><i>Der Algorithmus von Hierholzer.</i></a> In: <i>Einführung in die Mathematik 2.1 – Elementare Zahlentheorie und Graphentheorie.</i> 6. Oktober 2022.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-12-03" href="https://de.wikipedia.org/wiki/?title=Algorithmus_von_Hierholzer&oldid=250921613">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>