Graph embedding: Difference between revisions

Revision as of 12:42, 3 March 2023

Latest revision as of 19:55, 12 October 2024

(6 intermediate revisions by 5 users not shown)

Line 10:

* 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

Line 50:

==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.

Line 63:

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>