Graph embedding

<h2 id="terminology">Terminology</h2>
If a graph 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is embedded on a closed surface 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
, the complement of the union of the points and arcs associated with
the vertices and edges of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a family of regions (or <a href="/facts/Face_(graph_theory)/W0MqKkyE">faces</a>).<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> A 2-cell embedding, cellular embedding or map is an embedding in which every face is homeomorphic to an open disk.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> A closed 2-cell embedding is an embedding in which the closure of every face is homeomorphic to a closed disk.
The genus of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> is the minimal integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that the graph can be embedded in a surface of <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
. In particular, a <a href="/facts/Planar_graph/plREagr9">planar graph</a> has genus 
 
 
 
 0
 
 
 {\displaystyle 0}
 
, because it can be drawn on a sphere without self-crossing. A graph that can be embedded on a <a href="/facts/Torus/J9SkaV1W">torus</a> is called a <a href="/facts/Toroidal_graph/jtgbddDj">toroidal graph</a>.
The non-orientable genus of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> is the minimal integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that the graph can be embedded in a non-orientable surface of (non-orientable) genus 
 
 
 
 n
 
 
 {\displaystyle n}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
The Euler genus of a graph is the minimal integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that the graph can be embedded in an orientable surface of (orientable) genus 
 
 
 
 n
 
 /
 
 2
 
 
 {\displaystyle n/2}
 
 or in a non-orientable surface of (non-orientable) genus 
 
 
 
 n
 
 
 {\displaystyle n}
 
. A graph is orientably simple if its Euler genus is smaller than its non-orientable genus.
The maximum genus of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> is the maximal integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 such that the graph can be 
 
 
 
 2
 
 
 {\displaystyle 2}
 
-cell embedded in an orientable surface of <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
.

<h2 id="combinatorial-embedding">Combinatorial embedding</h2>
Main article: <a href="/facts/Rotation_system/9AaoWJ5a">Rotation system</a>
An embedded graph uniquely defines <a href="/facts/Cyclic_order/j5DaU26j">cyclic orders</a> of edges incident to the same vertex. The set of all these cyclic orders is called a <a href="/facts/Rotation_system/9AaoWJ5a">rotation system</a>. Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous definition in terms of points and curves). Sometimes, the rotation system itself is called a "combinatorial embedding".<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
An embedded graph also defines natural cyclic orders of edges which constitutes the boundaries of the faces of the embedding. However handling these face-based orders is less straightforward, since in some cases some edges may be traversed twice along a face boundary. For example this is always the case for embeddings of trees, which have a single face. To overcome this combinatorial nuisance, one may consider that every edge is "split" lengthwise in two "half-edges", or "sides". Under this convention in all face boundary traversals each half-edge is traversed only once and the two half-edges of the same edge are always traversed in opposite directions.
Other equivalent representations for cellular embeddings include the <a href="/facts/Ribbon_graph/z3rVR4oX">ribbon graph</a>, a topological space formed by gluing together topological disks for the vertices and edges of an embedded graph, and the <a href="/facts/Graph-encoded_map/m097Vu43">graph-encoded map</a>, an edge-colored <a href="/facts/Cubic_graph/6M32WUIJ">cubic graph</a> with four vertices for each edge of the embedded graph.

<h2 id="computational-complexity">Computational complexity</h2>
The problem of finding the graph genus is <a href="/facts/NP-hard/mQeNPv4R">NP-hard</a> (the problem of determining whether an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-vertex graph has genus 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is <a href="/facts/NP-complete/NFPv56DU">NP-complete</a>).<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
At the same time, the graph genus problem is <a href="/facts/Fixed-parameter_tractability/RoyaJGgJ">fixed-parameter tractable</a>, i.e., <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a> algorithms are known to check whether a graph can be embedded into a surface of a given fixed genus as well as to find the embedding.
The first breakthrough in this respect happened in 1979, when algorithms of <a href="/facts/Time_complexity/77T62gmf">time complexity</a>
O(nO(g)) were independently submitted to the Annual <a href="/facts/ACM_Symposium_on_Theory_of_Computing/xsemn7kG">ACM Symposium on Theory of Computing</a>: one by I. Filotti and <a href="/facts/Gary_Miller_(computer_scientist)/IFfqHGPp">G.L. Miller</a> and another one by <a href="/facts/John_Reif/q0e7NWr9">John Reif</a>. Their approaches were quite different, but upon the suggestion of the program committee they presented a joint paper.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> However, <a href="/facts/Wendy_Myrvold/ByYCdRkG">Wendy Myrvold</a> and <a href="/facts/William_Lawrence_Kocay/PpGAlGtN">William Kocay</a> proved in 2011 that the algorithm given by Filotti, Miller and Reif was incorrect.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
In 1999 it was reported that the fixed-genus case can be solved in time <a href="/facts/Linear_time/77T62gmf">linear</a> in the graph size and <a href="/facts/Double_exponential_function/HdrTmY6p">doubly exponential</a> in the genus.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="embeddings-of-graphs-into-higher-dimensional-spaces">Embeddings of graphs into higher-dimensional spaces</h2>
It is known that any finite graph can be embedded into a three-dimensional space.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
One method for doing this is to place the points on any line in space and to draw the edges as curves each of which lies in a distinct <a href="/facts/Halfplane/AGknBgnw">halfplane</a>, with all halfplanes having that line as their common boundary. An embedding like this in which the edges are drawn on halfplanes is called a <a href="/facts/Book_embedding/2hz3yIP7">book embedding</a> of the graph. This <a href="/facts/Metaphor/xwsLaqJI">metaphor</a> comes from imagining that each of the planes where an edge is drawn is like a page of a book. It was observed that in fact several edges may be drawn in the same "page"; the book thickness of the graph is the minimum number of halfplanes needed for such a drawing.
Alternatively, any finite graph can be drawn with straight-line edges in three dimensions without crossings by placing its vertices in <a href="/facts/General_position/p0YpbkJv">general position</a> so that no four are coplanar. For instance, this may be achieved by placing the ith vertex at the point (i,i2,i3) of the <a href="/facts/Moment_curve/6meK4TPq">moment curve</a>.
An embedding of a graph into three-dimensional space in which no two of the cycles are topologically linked is called a <a href="/facts/Linkless_embedding/Wjy3PNff">linkless embedding</a>. A graph has a linkless embedding if and only if it does not have one of the seven graphs of the <a href="/facts/Petersen_family/NNrfHd93">Petersen family</a> as a <a href="/facts/Minor_(graph_theory)/qn5eI22T">minor</a>.

<h2 id="gallery">Gallery</h2>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Embedding/HKeVKc42">Embedding</a>, for other kinds of embeddings</li>
<li><a href="/facts/Book_thickness/2hz3yIP7">Book thickness</a></li>
<li><a href="/facts/Graph_thickness/o9lnEd1c">Graph thickness</a></li>
<li><a href="/facts/Doubly_connected_edge_list/0ILWPPkp">Doubly connected edge list</a>, a data structure to represent a graph embedding in the <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a></li>
<li><a href="/facts/Regular_map_(graph_theory)/4ahwy17x">Regular map (graph theory)</a></li>
<li><a href="/facts/F%C3%A1ry%27s_theorem/BVE1xG9y">Fáry's theorem</a>, which says that a straight line planar embedding of a planar graph is always possible.</li>
<li><a href="/facts/Triangulation_(geometry)/enYNhLNy">Triangulation (geometry)</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Cohen, Robert F.; Eades, Peter; Lin, Tao; Ruskey, Frank (1995), "Three-dimensional graph drawing", in Tamassia, Roberto; Tollis, Ioannis G. (eds.), Graph Drawing: DIMACS International Workshop, GD '94 Princeton, New Jersey, USA, October 10–12, 1994, Proceedings, Lecture Notes in Computer Science, vol. 894, Springer, pp. 1–11, doi:10.1007/3-540-58950-3_351, ISBN 978-3-540-58950-1. <a href="978-3-540-58950-1" target="_blank">978-3-540-58950-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Katoh, Naoki; Tanigawa, Shin-ichi (2007), "Enumerating Constrained Non-crossing Geometric Spanning Trees", Computing and Combinatorics, 13th Annual International Conference, COCOON 2007, Banff, Canada, July 16-19, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4598, Springer-Verlag, pp. 243–253, CiteSeerX 10.1.1.483.874, doi:10.1007/978-3-540-73545-8_25, ISBN 978-3-540-73544-1. <a href="978-3-540-73544-1" target="_blank">978-3-540-73544-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gross, Jonathan; Tucker, Thomas W. (2001), Topological Graph Theory, Dover Publications, ISBN 978-0-486-41741-7. <a href="978-0-486-41741-7" target="_blank">978-0-486-41741-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on Surfaces and their Applications, Springer-Verlag, ISBN 978-3-540-00203-1. <a href="978-3-540-00203-1" target="_blank">978-3-540-00203-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Gross, Jonathan; Tucker, Thomas W. (2001), Topological Graph Theory, Dover Publications, ISBN 978-0-486-41741-7. <a href="978-0-486-41741-7" target="_blank">978-0-486-41741-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Mutzel, Petra; Weiskircher, René (2000), "Computing Optimal Embeddings for Planar Graphs", Computing and Combinatorics, 6th Annual International Conference, COCOON 2000, Sydney, Australia, July 26–28, 2000, Proceedings, Lecture Notes in Computer Science, vol. 1858, Springer-Verlag, pp. 95–104, doi:10.1007/3-540-44968-X_10, ISBN 978-3-540-67787-1. <a href="978-3-540-67787-1" target="_blank">978-3-540-67787-1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Didjev, Hristo N. (1995), "On drawing a graph convexly in the plane", Graph Drawing, DIMACS International Workshop, GD '94, Princeton, New Jersey, USA, October 10–12, 1994, Proceedings, Lecture Notes in Computer Science, vol. 894, Springer-Verlag, pp. 76–83, doi:10.1007/3-540-58950-3_358, ISBN 978-3-540-58950-1. <a href="978-3-540-58950-1" target="_blank">978-3-540-58950-1</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Duncan, Christian; Goodrich, Michael T.; Kobourov, Stephen (2010), "Planar Drawings of Higher-Genus Graphs", Graph Drawing, 17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25, 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 45–56, arXiv:0908.1608, doi:10.1007/978-3-642-11805-0_7, ISBN 978-3-642-11804-3. <a href="978-3-642-11804-3" target="_blank">978-3-642-11804-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Thomassen, Carsten (1989), "The graph genus problem is NP-complete", Journal of Algorithms, 10 (4): 568–576, doi:10.1016/0196-6774(89)90006-0 <a href="/wiki/Carsten_Thomassen_(mathematician)" target="_blank">/wiki/Carsten_Thomassen_(mathematician)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Filotti, I. S.; Miller, Gary L.; Reif, John (1979), "On determining the genus of a graph in O(v O(g)) steps(Preliminary Report)", Proc. 11th Annu. ACM Symposium on Theory of Computing, pp. 27–37, doi:10.1145/800135.804395. <a href="/wiki/Gary_Miller_(computer_scientist)" target="_blank">/wiki/Gary_Miller_(computer_scientist)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Myrvold, Wendy; Kocay, William (March 1, 2011). "Errors in Graph Embedding Algorithms". Journal of Computer and System Sciences. 2 (77): 430–438. doi:10.1016/j.jcss.2010.06.002. <a href="/wiki/Wendy_Myrvold" target="_blank">/wiki/Wendy_Myrvold</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Mohar, Bojan (1999), "A linear time algorithm for embedding graphs in an arbitrary surface", SIAM Journal on Discrete Mathematics, 12 (1): 6–26, CiteSeerX 10.1.1.97.9588, doi:10.1137/S089548019529248X <a href="/wiki/Bojan_Mohar" target="_blank">/wiki/Bojan_Mohar</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Cohen, Robert F.; Eades, Peter; Lin, Tao; Ruskey, Frank (1995), "Three-dimensional graph drawing", in Tamassia, Roberto; Tollis, Ioannis G. (eds.), Graph Drawing: DIMACS International Workshop, GD '94 Princeton, New Jersey, USA, October 10–12, 1994, Proceedings, Lecture Notes in Computer Science, vol. 894, Springer, pp. 1–11, doi:10.1007/3-540-58950-3_351, ISBN 978-3-540-58950-1. <a href="978-3-540-58950-1" target="_blank">978-3-540-58950-1</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

Graph embedding open-in-new

Graph embedding