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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>

<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>

<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>

<h3 id="eulers-theorem">Euler's theorem<a class="headerlink" href="#eulers-theorem" title="Permanent link">&para;</a></h3>

<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>

<div class="arithmatex">$$n - m + f = 2$$</div>