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‎main/algebra/all-submasks.html

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‎main/combinatorics/binomial-coefficients.html

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‎main/dynamic_programming/intro-to-dp.html

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‎main/feed_json_updated.json

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‎main/game_theory/sprague-grundy-nim.html

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‎main/geometry/nearest_points.html

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<li><a href="https://uva.onlinejudge.org/index.php?option=onlinejudge&amp;page=show_problem&amp;problem=1186">UVA 10245 "The Closest Pair Problem" [difficulty: low]</a></li>
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<li><a href="https://www.spoj.com/problems/CLOPPAIR/">SPOJ #8725 CLOPPAIR "Closest Point Pair" [difficulty: low]</a></li>
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<li><a href="http://codeforces.com/contest/120/problem/J">CODEFORCES Team Olympiad Saratov - 2011 "Minimum amount" [difficulty: medium]</a></li>
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<li><a href="https://code.google.com/codejam/contest/311101/dashboard#s=a&amp;a=1">Google CodeJam 2009 Final " Min Perimeter "[difficulty: medium]</a></li>
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<li><a href="https://github.com/google/coding-competitions-archive/blob/main/codejam/2009/world_finals/min_perimeter/statement.pdf">Google CodeJam 2009 Final "Min Perimeter" [difficulty: medium]</a></li>
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<li><a href="https://www.spoj.com/problems/CLOSEST/">SPOJ #7029 CLOSEST "Closest Triple" [difficulty: medium]</a></li>
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<li><a href="https://acm.timus.ru/problem.aspx?space=1&amp;num=1514">TIMUS 1514 National Park [difficulty: medium]</a></li>
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</ul>
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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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<span class="contributors-text">Contributors:</span>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/singamandeep" title="singamandeep" data-bi-name="contributorprofile" target="_blank">singamandeep</a> (92.13%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (3.93%)</li><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (2.81%)</li><li><a href="https://github.com/SYury" title="SYury" data-bi-name="contributorprofile" target="_blank">SYury</a> (0.56%)</li><li><a href="https://github.com/avishekp4" title="avishekp4" data-bi-name="contributorprofile" target="_blank">avishekp4</a> (0.56%)</li></ul>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/singamandeep" title="singamandeep" data-bi-name="contributorprofile" target="_blank">singamandeep</a> (91.57%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (3.93%)</li><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (2.81%)</li><li><a href="https://github.com/Kostero" title="Kostero" data-bi-name="contributorprofile" target="_blank">Kostero</a> (0.56%)</li><li><a href="https://github.com/SYury" title="SYury" data-bi-name="contributorprofile" target="_blank">SYury</a> (0.56%)</li><li><a href="https://github.com/avishekp4" title="avishekp4" data-bi-name="contributorprofile" target="_blank">avishekp4</a> (0.56%)</li></ul>
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‎main/geometry/planar.html

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Such graphs are called <strong>planar</strong>. Now suppose that we are given a planar graph together with its straight-line embedding, which means that for each vertex <span class="arithmatex">$v$</span> we have a corresponding point <span class="arithmatex">$(x, y)$</span> and all edges are drawn as line segments between these points without intersection (such embedding always exists). These line segments split the plane into several regions, which are called faces. Exactly one of the faces is unbounded. This face is called <strong>outer</strong>, while the other faces are called <strong>inner</strong>.</p>
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<p>In this article we will deal with finding both inner and outer faces of a planar graph. We will assume that the graph is connected.</p>
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<h2 id="some-facts-about-planar-graphs">Some facts about planar graphs<a class="headerlink" href="#some-facts-about-planar-graphs" title="Permanent link">&para;</a></h2>
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<p>In this section we present several facts about planar graphs without proof. Readers who are interested in proofs should refer to <a href="https://sites.math.washington.edu/~billey/classes/562.winter.2018/articles/GraphTheory.pdf">Graph Theory by R. Diestel</a> or some other book.</p>
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<p>In this section we present several facts about planar graphs without proof. Readers who are interested in proofs should refer to <a href="https://diestel-graph-theory.com/">Graph Theory by R. Diestel</a> (see <a href="https://www.youtube.com/@DiestelGraphTheory">video lectures</a> based on this book) or some other book.</p>
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<h3 id="eulers-theorem">Euler's theorem<a class="headerlink" href="#eulers-theorem" title="Permanent link">&para;</a></h3>
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<p>Euler's theorem states that any correct embedding of a connected planar graph with <span class="arithmatex">$n$</span> vertices, <span class="arithmatex">$m$</span> edges and <span class="arithmatex">$f$</span> faces satisfies:</p>
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<div class="arithmatex">$$n - m + f = 2$$</div>
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<span class="contributors-text">Contributors:</span>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/SYury" title="SYury" data-bi-name="contributorprofile" target="_blank">SYury</a> (99.72%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (0.28%)</li></ul>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/SYury" title="SYury" data-bi-name="contributorprofile" target="_blank">SYury</a> (99.43%)</li><li><a href="https://github.com/Kostero" title="Kostero" data-bi-name="contributorprofile" target="_blank">Kostero</a> (0.28%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (0.28%)</li></ul>
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‎main/graph/2SAT.html

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