Graph embedding: Difference between revisions

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* 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]]{{citation needed|date=August 2022}}, [[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

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the vertices and edges of <math>G</math> is a family of '''regions''' (or '''[[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=978-3-540-00203-1}}.</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. A graph that can be embedded on a [[torus]] is called a [[toroidal graph]].

The '''non-orientable genus''' of a [[Graph (discrete mathematics)|graph]] is the minimal integer <math>n</math> such that the graph can be embedded in a non-orientable surface of (non-orientable) genus <math>n</math>.<ref name="gt01"/>

The '''Euler genus''' of a graph is the minimal integer <math>n</math> such that the graph can be embedded in an orientable surface of (orientable) genus <math>n/2</math> or in a non-orientable surface of (non-orientable) genus <math>n</math>. A graph is '''orientably simple''' if its Euler genus is smaller than its non-orientable genus.

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==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=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=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.

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==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=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|pagepages=430–438|doi=10.1016/j.jcss.2010.06.002|doi-access=free}}</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>