1-planar graph
File:3-crossing Heawood graph.svg: six of the edges have a single crossing, and the remaining 15 edges are not crossed.]]
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of the graph.
Coloring
1-planar graphs were first studied by {{harvtxt|Ringel|1965}}, who showed that they can be colored with at most seven colors.{{citation
| last = Ringel | first = Gerhard | authorlink = Gerhard Ringel
| doi = 10.1007/BF02996313
| journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
| language = German
| mr = 0187232
| pages = 107–117
| title = Ein Sechsfarbenproblem auf der Kugel
| volume = 29
| year = 1965| issue = 1–2 | s2cid = 123286264 }}. Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six.{{citation
| last = Borodin | first = O. V.
| issue = 41
| journal = Metody Diskretnogo Analiza
| mr = 832128
| pages = 12–26, 108
| title = Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs
| year = 1984}}. The example of the complete graph K6, which is 1-planar, shows that 1-planar graphs may sometimes require six colors. However, the proof that six colors are always enough is more complicated.
Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and faces of a planar graph in such a way that no two adjacent vertices have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously be done using eight colors by applying the four color theorem to the given graph and its dual graph separately, using two disjoint sets of four colors. However, fewer colors may be obtained by forming an auxiliary graph that has a vertex for each vertex or face of the given planar graph, and in which two auxiliary graph vertices are adjacent whenever they correspond to adjacent features of the given planar graph. A vertex coloring of the auxiliary graph corresponds to a vertex-face coloring of the original planar graph. This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors. The graph K6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a triangular prism, then its eleven vertices and faces require six colors, because no three of them may be given a single color.{{citation
| last1 = Albertson | first1 = Michael O.
| last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar
| doi = 10.1007/s00373-006-0653-4
| issue = 3
| journal = Graphs and Combinatorics
| mr = 2264852
| pages = 289–295
| title = Coloring vertices and faces of locally planar graphs
| url = http://www.fmf.uni-lj.si/~mohar/Reprints/2006/BM06_GC22_Albertson_ColoringPlanarGraphs.pdf
| volume = 22
| year = 2006| s2cid = 1028234
}}.
Edge density
Every 1-planar graph with n vertices has at most 4n − 8 edges.{{citation
| last = Schumacher | first = H.
| journal = Mathematische Nachrichten
| language = German
| mr = 847368
| pages = 291–300
| title = Zur Struktur 1-planarer Graphen
| volume = 125
| year = 1986| doi = 10.1002/mana.19861250122
}}. More strongly, each 1-planar drawing has at most n − 2 crossings; removing one edge from each crossing pair of edges leaves a planar graph, which can have at most 3n − 6 edges, from which the 4n − 8 bound on the number of edges in the original 1-planar graph immediately follows.{{citation|title=On drawings and decompositions of 1-planar graphs|first1=Július|last1=Czap|first2=Dávid|last2=Hudák|journal=Electronic Journal of Combinatorics|year=2013|volume=20|issue=2|doi=10.37236/2392|at=P54|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p54|doi-access=free}}. However, unlike planar graphs (for which all maximal planar graphs on a given vertex set have the same number of edges as each other), there exist maximal 1-planar graphs (graphs to which no additional edges can be added while preserving 1-planarity) that have significantly fewer than 4n − 8 edges.{{citation
| last1 = Brandenburg | first1 = Franz Josef
| last2 = Eppstein | first2 = David | author2-link = David Eppstein
| last3 = Gleißner | first3 = Andreas
| last4 = Goodrich | first4 = Michael T. | author4-link = Michael T. Goodrich
| last5 = Hanauer | first5 = Kathrin
| last6 = Reislhuber | first6 = Josef
| editor1-last = Didimo | editor1-first = Walter
| editor2-last = Patrignani | editor2-first = Maurizio
| contribution = On the density of maximal 1-planar graphs
| title = Proc. 20th Int. Symp. Graph Drawing
| year = 2013}}. The bound of 4n − 8 on the maximum possible number of edges in a 1-planar graph can be used to show that the complete graph K7 on seven vertices is not 1-planar, because this graph has 21 edges and in this case 4n − 8 = 20 < 21.
A 1-planar graph is said to be an optimal 1-planar graph if it has exactly 4n − 8 edges, the maximum possible. In a 1-planar embedding of an optimal 1-planar graph, the uncrossed edges necessarily form a quadrangulation (a polyhedral graph in which every face is a quadrilateral). Every quadrangulation gives rise to an optimal 1-planar graph in this way, by adding the two diagonals to each of its quadrilateral faces. It follows that every optimal 1-planar graph is Eulerian (all of its vertices have even degree), that the minimum degree in such a graph is six, and that every optimal 1-planar graph has at least eight vertices of degree exactly six. Additionally, every optimal 1-planar graph is 4-vertex-connected, and every 4-vertex cut in such a graph is a separating cycle in the underlying quadrangulation.{{citation
| last = Suzuki | first = Yusuke
| doi = 10.1137/090746835
| issue = 4
| journal = SIAM Journal on Discrete Mathematics
| mr = 2746706
| pages = 1527–1540
| title = Re-embeddings of maximum 1-planar graphs
| volume = 24
| year = 2010}}.
The graphs that have straight 1-planar drawings (that is, drawings in which each edge is represented by a line segment, and in which each line segment is crossed by at most one other edge) have a slightly tighter bound of 4n − 9 on the maximum number of edges, achieved by infinitely many graphs.{{citation
| last = Didimo | first = Walter
| doi = 10.1016/j.ipl.2013.01.013
| issue = 7
| journal = Information Processing Letters
| mr = 3017985
| pages = 236–240
| title = Density of straight-line 1-planar graph drawings
| volume = 113
| year = 2013}}.
Complete multipartite graphs
File:1planar-K2222.svg K2,2,2,2]]
A complete classification of the 1-planar complete graphs, complete bipartite graphs, and more generally complete multipartite graphs is known. Every complete bipartite graph of the form K2,n is 1-planar (even planar), as is every complete tripartite graph of the form K1,1,n. Other than these infinite sets of examples, the only complete multipartite 1-planar graphs are K6, K1,1,1,6, K1,1,2,3, K2,2,2,2, K1,1,1,2,2, and their subgraphs. The minimal non-1-planar complete multipartite graphs are K3,7, K4,5, K1,3,4, K2,3,3, and K1,1,1,1,3.
For instance, the complete bipartite graph K3,6 is 1-planar because it is a subgraph of K1,1,1,6, but K3,7 is not 1-planar.{{citation
| last1 = Czap | first1 = Július
| last2 = Hudák | first2 = Dávid
| doi = 10.1016/j.dam.2011.11.014
| issue = 4–5
| journal = Discrete Applied Mathematics
| mr = 2876333
| pages = 505–512
| title = 1-planarity of complete multipartite graphs
| volume = 160
| year = 2012| doi-access =
}}.
Computational complexity
It is NP-complete to test whether a given graph is 1-planar,{{citation
| last1 = Grigoriev | first1 = Alexander
| last2 = Bodlaender | first2 = Hans L. | author2-link = Hans L. Bodlaender
| doi = 10.1007/s00453-007-0010-x
| issue = 1
| journal = Algorithmica
| mr = 2344391
| pages = 1–11
| title = Algorithms for graphs embeddable with few crossings per edge
| volume = 49
| year = 2007| hdl = 1874/17980
| s2cid = 8174422
| url = https://cris.maastrichtuniversity.nl/en/publications/5984fed8-f0b5-4b0d-91fe-2ca15d158421
| hdl-access = free
}}.{{citation
| last1 = Korzhik | first1 = Vladimir P.
| last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar
| editor1-last = Tollis | editor1-first = Ioannis G.
| editor2-last = Patrignani | editor2-first = Maurizio
| contribution = Minimal obstructions for 1-immersions and hardness of 1-planarity testing
| doi = 10.1007/978-3-642-00219-9_29
| pages = 302–312
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Graph Drawing: 16th International Symposium, GD 2008, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers
| volume = 5417
| year = 2009| arxiv = 1110.4881| isbn = 978-3-642-00218-2
| s2cid = 13436158
}}. and it remains NP-complete even for the graphs formed from planar graphs by adding a single edge{{citation | first1 = Sergio | last1 = Cabello | last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar | title = Adding one edge to planar graphs makes crossing number and 1-planarity hard | year = 2012 | arxiv = 1203.5944| bibcode = 2012arXiv1203.5944C }}. Expanded version of a paper from the 17th ACM Symposium on Computational Geometry, 2010. and for graphs of bounded bandwidth.{{citation|contribution=Parameterized complexity of 1-planarity|first1=Michael J.|last1=Bannister|first2=Sergio|last2=Cabello|first3=David|last3=Eppstein|author3-link=David Eppstein|title=Algorithms and Data Structures Symposium (WADS 2013)|year=2013|volume=22 |pages=23–49 |doi=10.7155/jgaa.00457|arxiv=1304.5591|bibcode=2013arXiv1304.5591B|s2cid=4417962}}. The problem is fixed-parameter tractable when parameterized by cyclomatic number or by tree-depth, so it may be solved in polynomial time when those parameters are bounded.
In contrast to Fáry's theorem for planar graphs, not every 1-planar graph may be drawn 1-planarly with straight line segments for its edges.{{citation
| last = Eggleton | first = Roger B.
| journal = Utilitas Mathematica
| mr = 846198
| pages = 149–172
| title = Rectilinear drawings of graphs
| volume = 29
| year = 1986}}.{{citation
| last = Thomassen | first = Carsten | authorlink = Carsten Thomassen (mathematician)
| doi = 10.1002/jgt.3190120306
| issue = 3
| journal = Journal of Graph Theory
| mr = 956195
| pages = 335–341
| title = Rectilinear drawings of graphs
| volume = 12
| year = 1988}}. However, testing whether a 1-planar drawing may be straightened in this way can be done in polynomial time.{{citation
| last1 = Hong | first1 = Seok-Hee | author1-link = Seok-Hee Hong
| last2 = Eades | first2 = Peter | author2-link = Peter Eades
| last3 = Liotta | first3 = Giuseppe
| last4 = Poon | first4 = Sheung-Hung
| editor1-last = Gudmundsson | editor1-first = Joachim
| editor2-last = Mestre | editor2-first = Julián
| editor3-last = Viglas | editor3-first = Taso
| contribution = Fáry's theorem for 1-planar graphs
| doi = 10.1007/978-3-642-32241-9_29
| pages = 335–346
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Computing and Combinatorics: 18th Annual International Conference, COCOON 2012, Sydney, Australia, August 20-22, 2012, Proceedings
| volume = 7434
| year = 2012| isbn = 978-3-642-32240-2 }}. Additionally, every 3-vertex-connected 1-planar graph has a 1-planar drawing in which at most one edge, on the outer face of the drawing, has a bend in it. This drawing can be constructed in linear time from a 1-planar embedding of the graph.{{citation
| last1 = Alam | first1 = Md. Jawaherul
| last2 = Brandenburg | first2 = Franz J.
| last3 = Kobourov | first3 = Stephen G.
| contribution = Straight-line grid drawings of 3-connected 1-planar graphs
| doi = 10.1007/978-3-319-03841-4_8
| pages = 83–94
| series = Lecture Notes in Computer Science
| title = Graph Drawing: 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers
| url = http://www.cs.arizona.edu/~mjalam/straight_1_planar_3_connected.pdf
| volume = 8242
| year = 2013| doi-access = free
| isbn = 978-3-319-03840-7
}}. The 1-planar graphs have bounded book thickness,{{citation
| last1 = Bekos | first1 = Michael A.
| last2 = Bruckdorfer | first2 = Till
| last3 = Kaufmann | first3 = Michael
| last4 = Raftopoulou | first4 = Chrysanthi
| contribution = 1-Planar graphs have constant book thickness
| doi = 10.1007/978-3-662-48350-3_12
| pages = 130–141
| publisher = Springer
| series = Lecture Notes in Computer Science
| title = Algorithms – ESA 2015
| volume = 9294
| year = 2015| isbn = 978-3-662-48349-7
}}. but some 1-planar graphs including K2,2,2,2 have book thickness at least four.{{citation|contribution=The book embedding problem from a SAT-solving perspective|first1=Michael|last1=Bekos|first2=Michael|last2=Kaufmann|first3=Christian|last3=Zielke|pages=113–125|title=Proc. 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015)|year=2015}}.
1-planar graphs have bounded local treewidth, meaning that there is a (linear) function f such that the 1-planar graphs of diameter d have treewidth at most f(d); the same property holds more generally for the graphs that can be embedded onto a surface of bounded genus with a bounded number of crossings per edge. They also have separators, small sets of vertices the removal of which decomposes the graph into connected components whose size is a constant fraction of the size of the whole graph. Based on these properties, numerous algorithms for planar graphs, such as Baker's technique for designing approximation algorithms, can be extended to 1-planar graphs. For instance, this method leads to a polynomial-time approximation scheme for the maximum independent set of a 1-planar graph.{{harvtxt|Grigoriev|Bodlaender|2007}}. Grigoriev and Bodlaender state their results only for graphs with a known 1-planar embedding, and use a tree decomposition of a planarization of the embedding with crossings replaced by degree-four vertices; however, their methods straightforwardly imply bounded local treewidth of the original 1-planar graph, allowing Baker's method to be applied directly to it without knowing the embedding.
References
{{reflist|30em}}
Further reading
- {{citation
| last1 = Kobourov | first1 = Stephen
| last2 = Liotta | first2 = Giuseppe
| last3 = Montecchiani | first3 = Fabrizio
| arxiv = 1703.02261
| title = An annotated bibliography on 1-planarity
| journal = Computer Science Review
| year = 2017| volume = 25
| pages = 49–67
| doi = 10.1016/j.cosrev.2017.06.002
| bibcode = 2017arXiv170302261K| s2cid = 7732463
}}