Graph embedding: Difference between revisions

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{{Use American English|date = February 2019}}

In [[topological graph theory]], an '''embedding''' (also spelled '''imbedding''') of a [[Graph (discrete mathematics)|graph]] <math>G</math> on a [[surface (mathematics)|surface]] Σ is a representation of <math>G</math> on Σ in which points of Σ are associated to [[graph theory|vertices]] and simple arcs ([[Homeomorphism|homeomorphic]] images of [0,1]) are associated to [[graph theory|edges]] in such a way that:

{{Short description|Embedding a graph in a topological space, often Euclidean}}

* the endpoints of the arc associated to an edge <math>e</math> are the points associated to the end vertices of <math>e</math>,

{{Use mdy dates|date = February 2019}}

[[File:Heawood graph and map on torus.png|thumb|The [[Heawood graph]] and associated map embedded in the torus.]]

In [[topological graph theory]], an '''embedding''' (also spelled '''imbedding''') of a [[Graph (discrete mathematics)|graph]] <math>G</math> on a [[surface (mathematics)|surface]] Σ<math>\Sigma</math> is a representation of <math>G</math> on Σ<math>\Sigma</math> in which points of Σ<math>\Sigma</math> are associated towith [[graph theory|vertices]] and simple arcs ([[Homeomorphism|homeomorphic]] images of <math>[0,1]</math>) are associated towith [[graph theory|edges]] in such a way that:

* the endpoints of the arc associated towith an edge <math>e</math> are the points associated towith the end vertices of <math>e,</math>,

* no arcs include points associated with other vertices,

* two arcs never intersect at a point which is interior to either of the arcs.

Here a surface is a [[compact space|compact]], [[connected space|connected]] <math>2</math>-[[manifold]].

Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space <math>\mathbb{R}^3</math>.<ref name="3d-gd">{{citation

| last1 = Cohen | first1 = Robert F.

| last2 = Eades | first2 = Peter | author2-link = Peter Eades

Line 19 ⟶ 26:

| title = Graph Drawing: DIMACS International Workshop, GD '94 Princeton, New Jersey, USA, October 10–12, 1994, Proceedings

| volume = 894

| year = 1995| isbn = 978-3-540-58950-1
| doi-access = free
}}.</ref> andA [[planar graphsgraph]] is one that can be embedded in 2-dimensional Euclidean space <math>\mathbb{R}^2.</math>.

Often, an '''embedding''' is regarded as an equivalence class (under homeomorphisms of Σ<math>\Sigma</math>) of representations of the kind just described.

Some authors define a weaker version of the definition of "graph embedding" by omitting the non-intersection condition for edges. In such contexts the stricter definition is described as "non-crossing graph embedding".<ref>{{citation|first1=Naoki|last1=Katoh|first2=Shin-ichi|last2=Tanigawa|title=Computing and Combinatorics, 13th Annual International Conference, COCOON 2007, Banff, Canada, July 16-19, 2007, Proceedings|publisher=Springer-Verlag|series=[[Lecture Notes in Computer Science]]|volume=4598|year=2007|pages=243–253|doi=10.1007/978-3-540-73545-8_25|chapter=Enumerating Constrained Non-crossing Geometric Spanning Trees|isbn=978-3-540-73544-1|citeseerx=10.1.1.483.874}}.</ref>

This article deals only with the strict definition of graph embedding. The weaker definition is discussed in the articles "[[graph drawing]]" and "[[Crossing number (graph theory)|crossing number]]".

==Terminology==

If a graph <math>G</math> is embedded on a closed surface Σ<math>\Sigma</math>, the complement of the union of the points and arcs associated towith

the vertices and edges of <math>G</math> is a family of '''regions''' (or '''faces[[face (graph theory)|face]]s''').<ref name="gt01">{{citation|last1=Gross|first1=Jonathan|last2=Tucker|first2=Thomas W.|authorlink2= Thomas W. Tucker| title=Topological Graph Theory|publisher=Dover Publications|year=2001|isbn=978-0-486-41741-7}}.</ref> A '''2-cell embedding''', '''cellular embedding''' or '''map''' is an embedding in which every face is homeomorphic to an open disk.<ref>{{citation|last1=Lando|first1=Sergei K.|last2=Zvonkin|first2=Alexander K.|title=Graphs on Surfaces and their Applications|publisher=Springer-Verlag|year=2004|isbn=978-3-540-00203-01}}.</ref> 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 [[Graph (discrete mathematics)|graph]] is the minimal integer ''<math>n''</math> such that the graph can be embedded in a surface of [[Genus (mathematics)|genus]] ''<math>n''</math>. In particular, a [[planar graph]] has genus <math>0</math>, because it can be drawn on a sphere without self-crossing. The '''non-orientableA genus''' of a [[Graph (discrete mathematics)|graph]] is the minimal integer ''n'' such that the graph can be embedded inon a non-orientable[[torus]] surfaceis ofcalled (non-orientable)a genus[[toroidal ''n''graph]].<ref name="gt01"/>

The '''Eulernon-orientable genus''' of a [[Graph (discrete mathematics)|graph]] is the minimal integer ''<math>n''</math> such that the graph can be embedded in an orientable surface of (orientable) genus ''n/2'' or in a non-orientable surface of (non-orientable) genus ''<math>n''</math>.<ref A graph is '''orientably simple''' if its Euler genus is smaller than its non-orientable genus.name="gt01"/>

The '''maximumEuler genus''' of a [[Graph (discrete mathematics)|graph]] is the maximalminimal integer ''<math>n''</math> such that the graph can be 2-cell embedded in an orientable surface of [[Genus(orientable) genus <math>n/2</math> or in a non-orientable surface of (mathematicsnon-orientable)| genus]] <math>n</math>. A graph is ''n'orientably simple''' if its Euler genus is smaller than its non-orientable genus.

The '''maximum genus''' of a [[Graph (discrete mathematics)|graph]] is the maximal integer <math>n</math> such that the graph can be <math>2</math>-cell embedded in an orientable surface of [[Genus (mathematics)|genus]] <math>n</math>.

==Combinatorial embedding==

{{main|Rotation system}}

An embedded graph uniquely defines [[cyclic order|cyclic orders]]s of edges incident to the same vertex. The set of all these cyclic orders is called a [[rotation system]]. 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".<ref>{{citation|first1=Petra|last1=Mutzel|author1-link=Petra Mutzel|first2=René|last2=Weiskircher|title=Computing and Combinatorics, 6th Annual International Conference, COCOON 2000, Sydney, Australia, July 26–28, 2000, Proceedings|year=2000|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=1858|pages=95–104|doi=10.1007/3-540-44968-X_10|chapter=Computing Optimal Embeddings for Planar Graphs|isbn=978-3-540-67787-1}}.</ref><ref>{{citation|last=Didjev|first=Hristo N.|contribution=On drawing a graph convexly in the plane|title=[[International Symposium on Graph Drawing|Graph Drawing, DIMACS International Workshop, GD '94, Princeton, New Jersey, USA, October 10–12, 1994, Proceedings]]|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|year=1995|volume=894|pages=76–83|doi=10.1007/3-540-58950-3_358|title-link=International Symposium on Graph Drawing|isbn=978-3-540-58950-1|doi-access=free}}.</ref><ref>{{citation|first1=Christian|last1=Duncan|first2=Michael T.|last2=Goodrich|author2-link=Michael T. Goodrich|first3=Stephen|last3=Kobourov|title=[[International Symposium on Graph Drawing|Graph Drawing, 17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25, 2009, Revised Papers]]|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|year=2010|pages=45–56|doi=10.1007/978-3-642-11805-0_7|volume=5849|chapter=Planar Drawings of Higher-Genus Graphs|isbn=978-3-642-11804-3|title-link=International Symposium on Graph Drawing|arxiv=0908.1608}}.</ref>

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 [[ribbon graph]], a topological space formed by gluing together topological disks for the vertices and edges of an embedded graph, and the [[graph-encoded map]], an edge-colored [[cubic graph]] with four vertices for each edge of the embedded graph.

==Computational complexity==

The problem of finding the graph genus is [[NP-hard]] (the problem of determining whether an ''<math>n''</math>-vertex graph has genus ''<math>g''</math> is [[NP-complete]]).<ref>{{Citation | last1=Thomassen | first1=Carsten | authorlink = Carsten Thomassen (mathematician) | title=The graph genus problem is NP-complete | year=1989 | journal=Journal of Algorithms | volume=10 | issue = 4 | pages=568–576 | doi = 10.1016/0196-6774(89)90006-0}}</ref>

At the same time, the graph genus problem is [[Fixed-parameter tractability|fixed-parameter tractable]], i.e., [[polynomial time]] 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 [[time complexity]]

''O''(''n''^''O''(''g'')) were independently submitted to the Annual [[ACM Symposium on Theory of Computing]]: one by I. Filotti and [[Gary Miller (computer scientist)|G.L. Miller]] and another one by [[John Reif]]. Their approaches were quite different, but upon the suggestion of the program committee they presented a joint paper.<ref>{{citation|first1=I. S.|last1=Filotti|author2-link=Gary Miller (computer scientist)|first2=Gary L.|last2=Miller|first3=John|last3=Reif|author3-link=John Reif |title=[[Symposium on Theory of Computing|Proc. 11th Annu. ACM Symposium on Theory of Computing]]|year=1979|pages=27–37|doi=10.1145/800135.804395|chapter=On determining the genus of a graph in O(v O(g)) steps(Preliminary Report)|title-link=Symposium on Theory of Computing|doi-access=free}}.</ref> However, [[Wendy Myrvold]] and [[William Lawrence Kocay|William Kocay]] proved in 2011 that the algorithm given by Filotti, Miller and Reif was incorrect.<ref>{{cite journal|last1=Myrvold|first1=Wendy|author1-link=Wendy Myrvold|last2=Kocay|first2=William|author2-link=William Lawrence Kocay|title=Errors in Graph Embedding Algorithms|journal=Journal of Computer and System Sciences|date=March 1, 2011|volume=2|issue=77|pages=430–438|doi=10.1016/j.jcss.2010.06.002|doi-access=}}</ref>

In 1999 it was reported that the fixed-genus case can be solved in time [[linear time|linear]] in the graph size and [[Double exponential function|doubly exponential]] in the genus.<ref>{{Citation | last1=Mohar | first1=Bojan | authorlink = Bojan Mohar | title=A linear time algorithm for embedding graphs in an arbitrary surface | year=1999 | journal=[[SIAM Journal on Discrete Mathematics]] | volume=12 | issue=1 | pages=6–26 | doi = 10.1137/S089548019529248X| citeseerx=10.1.1.97.9588 }}</ref>

==Embeddings of graphs into higher-dimensional spaces==

It is known that any finite graph can be embedded into a three-dimensional space.<ref name="3d-gd"/>

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 [[halfplane]], 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 [[book embedding]] of the graph. This [[metaphor]] 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 [[general position]] so that no four are coplanar. For instance, this may be achieved by placing the ''i''th vertex at the point (''i'',''i''²,''i''³) of the [[moment curve]].

An embedding of a graph into three-dimensional space in which no two of the cycles are topologically linked is called a [[linkless embedding]]. A graph has a linkless embedding if and only if it does not have one of the seven graphs of the [[Petersen family]] as a [[minor (graph theory)|minor]].

==Gallery==

<gallery>

File:Petersen-graph.png|The [[Petersen graph]] and associated map embedded in the projective plane. Opposite points on the circle are identified yielding a closed surface of non-orientable genus 1.

File:Pappus-graph-on-torus.png|The [[Pappus graph]] and associated map embedded in the torus.

File:Klein-map.png|The degree 7 [[Klein graph]] and associated map embedded in an orientable surface of genus 3.

</gallery>

==See also==

* [[Embedding]], for other kinds of embeddings

* [[Book thickness]]

* [[Geometric thickness]]

* [[Graph thickness]]

* [[Doubly connected edge list]], a data structure to represent a graph embedding in the [[plane (geometry)|plane]]