2076:
handles) and thus the genus of a graph is well defined. Obviously, if the graph can be embedded without crossings into a (orientable, connected, closed) surface with genus g, it can be embedded without crossings into all (orientable, connected, closed) surfaces with greater or equal genus. There are also other concepts in graph theory that are called "X genus" with "X" some qualifier; in general these differ from the above defined concept of "genus" without any qualifier. Especially the non-orientable genus of a graph (using non-orientable surfaces in its definition) is different for a general graph from the genus of that graph (using orientable surfaces in its definition).
1050:
242:
365:
953:
1357:
74:
63:
111:
1228:
93:
631:, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used.
2084:
where electrical connection between the sides of the board can be made (as is possible with typical real life circuit boards, with the electrical connections on the top side of the board achieved through pieces of wire and at the bottom side by tracks of copper constructed on to the board itself and
2079:
Any graph may be embedded into three-dimensional space without crossings. In fact, any graph can be drawn without crossings in a two plane setup, where two planes are placed on top of each other and the edges are allowed to "jump up" and "drop down" from one plane to the other at any place (not just
2089:
them into the tracks); one can also interpret this as saying that in order to build any road network, one only needs just bridges or just tunnels, not both (2 levels is enough, 3 is not needed). Also, in three dimensions the question about drawing the graph without crossings is trivial. However, a
2075:
of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Every graph can be embedded without crossings into some (orientable, connected) closed two-dimensional surface (sphere with
2059:
is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the
2060:
result can be nonplanar (for example, if one thinks of a circle divided into sectors, with the sectors being the regions, then the corresponding map graph is the complete graph as all the sectors have a common boundary point - the centre point).
1095:
that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges.
3126:— Free C source code for reference implementation of Boyer–Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator. An open source project with free licensing provides the
1076:) whenever they intersect in exactly one point. A "coin graph" is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The
1612:
planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs, as well as certain face subdivisions of triangular grid graphs, and certain triangulations of grid-covered cylinder graphs.
980:
of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example.
682:
states that a graph is planar if and only if it has a drawing in which each independent pair of edges crosses an even number of times; it can be used to characterize the planar graphs via a system of equations
149:, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a
1902:. Every simple outerplanar graph admits an embedding in the plane such that all vertices lie on a fixed circle and all edges are straight line segments that lie inside the disk and don't intersect, so
1438:
is a triangle. In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also the
1714:
1332:
Duals are useful because many properties of the dual graph are related in simple ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs.
900:, planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler's formula, one can then show that these graphs are
2080:
at the graph vertexes) so that the edges can avoid intersections with other edges. This can be interpreted as saying that it is possible to make any electrical conductor network with a two-sided
1562:
that can be drawn in the plane with its edges as non-crossing curves that are consistently oriented in an upward direction. Not every planar directed acyclic graph is upward planar, and it is
1372:
if it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are then bounded by three edges, explaining the alternative term
1538:
is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a
1020:, because each face has at least three face-edge incidences and each edge contributes exactly two incidences. It follows via algebraic transformations of this inequality with Euler's formula
1782:
1740:
1192:
441:
of a graph results from taking a subgraph and repeatedly contracting an edge into a vertex, with each neighbor of the original end-vertices becoming a neighbor of the new vertex.
1461:
are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true:
445:
947:
1862:
1835:
761:
1898:
vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of the
1808:
1648:
161:
on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
2245:
Buhl, J.; Gautrais, J.; Sole, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, G. (2004), "Efficiency and robustness in ant networks of galleries",
1470:
is planar but not outerplanar. A theorem similar to
Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of
2467:
1424:
are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar
655:
gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right
1601:
even states that for simple 3-vertex-connected planar graphs the position of the inner vertices can be chosen to be the average of its neighbors.
2112:
as a minor, the linklessly embeddable graphs may be characterized as the graphs that do not contain as a minor any of the seven graphs in the
1376:. The alternative names "triangular graph" or "triangulated graph" have also been used, but are ambiguous, as they more commonly refer to the
652:
157:
of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a
2998:
675:
gives a characterization based on the maximum multiplicity of the second eigenvalue of certain Schrödinger operators defined by the graph.
2098:
with each other. In analogy to
Kuratowski's and Wagner's characterizations of the planar graphs as being the graphs that do not contain
502:
In practice, it is difficult to use
Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast
3082:
3136:— GPL graph algorithm library including planarity testing, planarity embedder and Kuratowski subgraph exhibition in linear time.
3176:
2694:
Bonichon, N.; Gavoille, C.; Hanusse, N.; Poulalhon, D.; Schaeffer, G. (2006). "Planar Graphs, via Well-Orderly Maps and Trees".
2854:
Filotti, I. S.; Mayer, Jack N. (1980). "A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus".
2085:
electrical connection between the sides of the board achieved through drilling holes, passing the wires through the holes and
1653:
1035:
that for finite planar graphs the average degree is strictly less than 6. Graphs with higher average degree cannot be planar.
2876:
2229:
2190:
1542:
in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low
2555:
Chen, T. Z. Q.; Kitaev, S.; Sun, B. Y. (2016). "Word-representability of triangulations of grid-covered cylinder graphs".
2120:
at most two or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at most four.
2117:
672:
2333:
Bhasker, Jayaram; Sahni, Sartaj (1988), "A linear algorithm to find a rectangular dual of a planar triangulated graph",
2510:
Chen, T. Z. Q.; Kitaev, S.; Sun, B. Y. (2016). "Word-representability of face subdivisions of triangular grid graphs".
2420:
2156:, a pencil-and-paper game where a planar graph subject to certain constraints is constructed as part of the game play
1124:
641:
972:. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using the
3078:
2014:
Any planar graph on n nodes has at most 8(n-2) maximal cliques, which implies that the class of planar graphs is a
635:
294:
2814:
Dębski, Michał; Felsner, Stefan; Micek, Piotr; Schröder, Felix (2021), "Improved Bounds for
Centered Colorings",
1745:
1976:
of a graph of treewidth at most 8 and a path. This result has been used to show that planar graphs have bounded
28:
3142:, including linear time planarity testing, embedding, Kuratowski subgraph isolation, and straight-line drawing.
2204:
Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.
3021:
1936:
1719:
473:
17:
314:
279:
1145:
465:
138:
3139:
1913:
990:
891:
795:
faces, any change to the graph that creates an additional face while keeping the graph planar would keep
707:
1242:
of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the
1091:, that every simple planar graph can be embedded in the plane in such a way that its edges are straight
698:
2747:
1887:
1590:
354:
318:
2138:, a planar graph formed from a drawing with crossings by replacing each crossing point by a new vertex
722:
is the number of faces (regions bounded by edges, including the outer, infinitely large region), then
2008:
3123:
2708:
2116:. In analogy to the characterizations of the outerplanar and planar graphs as being the graphs with
3036:
2671:
2141:
1928:
1609:
1594:
1361:
679:
334:
191:
169:
1546:, making many algorithmic problems on them more easily solved than in unrestricted planar graphs.
3171:
2159:
2015:
917:
298:
257:
2094:, graphs that can be embedded into three-dimensional space in such a way that no two cycles are
1972:
The planar product structure theorem states that every planar graph is a subgraph of the strong
993:
simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are
468:
asked more generally whether any minor-closed class of graphs is determined by a finite set of "
3166:
2703:
2666:
2144:, the smallest number of planar graphs into which the edges of a given graph may be partitioned
1840:
1559:
1496:
by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.
1105:
1077:
1049:
1044:
977:
817:
1813:
728:
1555:
982:
662:
195:
3088:
2217:
1339:), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-
820:
it holds for all cases. Euler's formula can also be proved as follows: if the graph isn't a
3145:
2923:
2619:
2600:
Giménez, Omer; Noy, Marc (2009). "Asymptotic enumeration and limit laws of planar graphs".
2430:
2389:
2311:
2254:
2044:
is a graph that may be drawn in the plane with at most one simple crossing per edge, and a
1295:
is again the embedding of a (not necessarily simple) planar graph; it has as many edges as
884:
825:
438:
430:
404:
290:
2990:
1883:
1088:
8:
2072:
1891:
1627:
821:
400:
261:
245:
241:
217:
210:
2623:
2407:, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Wiesbaden, pp. 6–7,
2258:
3150:
3066:
3048:
2973:
2927:
2882:
2855:
2837:
2819:
2795:
2759:
2721:
2635:
2609:
2582:
2564:
2537:
2519:
2492:
2350:
2315:
2270:
2150:, a puzzle computer game in which the objective is to embed a planar graph onto a plane
2091:
1917:
1873:
1793:
1633:
1431:
1421:
357:
of a graph results from inserting vertices into edges (for example, changing an edge
3109:
2977:
2872:
2841:
2445:
Graph Theory: Proceedings of a
Conference held in Lagów, Poland, February 10–13, 1981
2416:
2319:
2225:
2186:
2180:
2129:
1458:
1072:
969:
656:
524:
187:
176:
3070:
2886:
2725:
2586:
2541:
2496:
2274:
3105:
3058:
3007:
2965:
2942:
2864:
2829:
2769:
2713:
2676:
2639:
2627:
2574:
2529:
2482:
2448:
2447:, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 248–256,
2408:
2377:
2354:
2342:
2299:
2262:
1582:. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph
1539:
1499:
A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For
1435:
986:
973:
957:
194:
drawings on the sphere, usually with additional assumptions such as the absence of
146:
2787:
2739:
2631:
2654:
2426:
2385:
2307:
2290:
2266:
2113:
1985:
1907:
1899:
1598:
767:
666:
469:
249:
225:
180:
142:
67:
2412:
2185:(Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64.
2680:
2153:
2095:
2064:
2041:
1877:
1579:
1447:
1381:
994:
512:
449:
322:
310:
97:
78:
3062:
2717:
2578:
2533:
2487:
2403:
Felsner, Stefan (2004), "1.4 Outerplanar Graphs and Convex
Geometric Graphs",
364:
3160:
2986:
2135:
2081:
1973:
1589:). A sufficient condition that a graph can be drawn convexly is that it is a
1439:
1385:
998:
346:
224: 0, since the plane (and the sphere) are surfaces of genus 0. See "
115:
32:
2947:
2908:
2368:
Seymour, P. D.; Weaver, R. W. (1984), "A generalization of chordal graphs",
2381:
2004:
1977:
1957:
1921:
1340:
1120:
1092:
1084:
in 1936, states that a graph is planar if and only if it is a coin graph.
961:
896:
597:
306:
130:
36:
27:"Triangular graph" redirects here. For line graphs of complete graphs, see
2868:
2033:-apex graph is a graph that may be made planar by the removal of at most
1981:
1563:
1535:
1336:
710:, planar graph is drawn in the plane without any edge intersections, and
645:
221:
158:
1916:(now a theorem) states that every planar graph can be represented as an
952:
3127:
3034:
3012:
2969:
2833:
2452:
2346:
2303:
2029:
is a graph that may be made planar by the removal of one vertex, and a
2026:
1443:
1377:
1356:
1335:
While the dual constructed for a particular embedding is unique (up to
1244:
1232:
1081:
644:
gives an algebraic characterization of finite planar graphs, via their
1787:
Almost all planar graphs have an exponential number of automorphisms.
476:, proved in a long series of papers. In the language of this theorem,
3053:
3025:
2743:
2614:
2147:
2086:
2056:
1988:
of near-linear size. It also has applications to vertex ranking and
1953:
1543:
1510:-outerplanar if removing the vertices on the outer face results in a
638:
gives a characterization based on the existence of an algebraic dual;
503:
2773:
1009:
Connected planar graphs with more than one edge obey the inequality
216:
Planar graphs generalize to graphs drawable on a surface of a given
73:
3096:
Fisk, Steve (1978), "A short proof of Chvátal's watchman theorem",
3087:(Technical report). UNM-ECE Technical Report 03-002. Archived from
2857:
Proceedings of the 12th Annual ACM Symposium on Theory of
Computing
2824:
2800:
2764:
2569:
2524:
2443:
Sysło, Maciej M.; Proskurowski, Andrzej (1983), "On Halin graphs",
2090:
three-dimensional analogue of the planar graphs is provided by the
1886:
states that every simple planar graph admits a representation as a
1388:
respectively. Every maximal planar graph is at least 3-connected.
62:
2991:"On the cutting edge: Simplified O(n) planarity by edge addition"
2899:
Wood, D. R. (2007). On the
Maximum Number of Cliques in a Graph.
850:
constant. Repeat until the remaining graph is a tree; trees have
110:
3133:
2956:
Wagner, K. (1937), "Ăśber eine
Eigenschaft der ebenen Komplexe",
2693:
494:
are the forbidden minors for the class of finite planar graphs.
1425:
1325:
is the planar graph corresponding to a convex polyhedron, then
1318:
286:
165:
1227:
791:. In general, if the property holds for all planar graphs of
3134:
2068:
444:
213:, none of the faces of a planar map has a particular status.
164:
Every graph that can be drawn on a plane can be drawn on the
2468:"Semi-transitive orientations and word-representable graphs"
1364:
is maximal planar. All its faces are bounded by three edges.
1317:; here the equality is the equivalence of embeddings on the
1307:
has vertices. The term "dual" is justified by the fact that
2785:
1488:. The above is a direct corollary of the fact that a graph
809:
an invariant. Since the property holds for all graphs with
3124:
Edge Addition Planarity Algorithm Source Code, version 1.0
2813:
2288:
Schnyder, W. (1989), "Planar graphs and poset dimension",
1790:
The number of unlabeled (non-isomorphic) planar graphs on
1329:
is the planar graph corresponding to the dual polyhedron.
1112:
of a planar graph, or network, is the ratio of the number
523:(linear time) whether the graph may be planar or not (see
92:
29:
Line graph § Strongly regular and perfect line graphs
3039:; Rosenstiehl, P. (2006), "Trémaux trees and planarity",
2465:
2067:
is a graph that can be embedded without crossings on the
411:
A finite graph is planar if and only if it does not have
293:
provided a characterization of planar graphs in terms of
3041:
International Journal of Foundations of Computer Science
2048:-planar graph is a graph that may be drawn with at most
1709:{\displaystyle g\cdot n^{-7/2}\cdot \gamma ^{n}\cdot n!}
2738:
2792:
Asymptotically optimal vertex ranking of planar graphs
2657:; Welsh, Dominic J.A. (2005). "Random planar graphs".
1952:) vertices. As a consequence, planar graphs also have
1062:, the complete graph on five vertices, minus one edge.
989:
formed from convex polyhedra are precisely the finite
35:. For data graphs plotted across three variables, see
2244:
1894:
is a set of points such that every planar graph with
1843:
1816:
1796:
1748:
1722:
1656:
1636:
1279:. Furthermore, this edge is drawn so that it crosses
1148:
920:
731:
236:
2652:
1212:
for a completely sparse planar graph (a tree), and
567:
Theorem 2. If there are no cycles of length 3, then
3128:
Edge Addition Planarity Algorithms, current version
2750:(2020), "Planar graphs have bounded queue number",
2132:
a combinatorial object that can encode plane graphs
1604:
1578:if all of its faces (including the outer face) are
1442:, and are exactly the graphs that can be formed by
2928:"Sur le problème des courbes gauches en topologie"
2466:HalldĂłrsson, M.; Kitaev, S.; Pyatkin., A. (2016).
1856:
1829:
1802:
1776:
1734:
1708:
1642:
1186:
964:, forming a planar graph from a convex polyhedron.
941:
755:
665:gives a characterization of planarity in terms of
2442:
1935:-vertex planar graph can be partitioned into two
1251:as follows: we choose one vertex in each face of
3158:
1999:vertices, it is possible to determine in time O(
1566:to test whether a given graph is upward planar.
542:faces, the following simple conditions hold for
386:subgraph. However, it contains a subdivision of
2459:
1621:
1219:for a completely dense (maximal) planar graph.
611:edges, asymptotically smaller than the maximum
3153:— NetLogo version of John Tantalo's game
3084:A New Parallel Algorithm for Planarity Testing
510:vertices, it is possible to determine in time
2554:
2509:
2367:
1630:for the number of (labeled) planar graphs on
1255:(including the outer face) and for each edge
3077:
2999:Journal of Graph Algorithms and Applications
2984:
2853:
2790:; Javarsineh, Mehrnoosh; Morin, Pat (2020),
2602:Journal of the American Mathematical Society
2332:
1346:
3140:Boost Graph Library tools for planar graphs
1777:{\displaystyle g\approx 0.43\times 10^{-5}}
530:For a simple, connected, planar graph with
3020:
2922:
2599:
2215:
1001:to a sphere, regardless of its convexity.
220:. In this terminology, planar graphs have
3098:Journal of Combinatorial Theory, Series B
3052:
3011:
2946:
2909:https://doi.org/10.1007/s00373-007-0738-8
2823:
2799:
2763:
2707:
2670:
2659:Journal of Combinatorial Theory, Series B
2613:
2568:
2548:
2523:
2486:
1066:We say that two circles drawn in a plane
1053:Example of the circle packing theorem on
653:Fraysseix–Rosenstiehl planarity criterion
2287:
1492:is outerplanar if the graph formed from
1355:
1351:
1226:
1048:
951:
824:, then remove an edge which completes a
699:Euler characteristic § Plane graphs
443:
363:
240:
2503:
2402:
2178:
1735:{\displaystyle \gamma \approx 27.22687}
1569:
1549:
1283:exactly once and that no other edge of
1099:
673:Colin de Verdière's planarity criterion
42:Graph that can be embedded in the plane
14:
3159:
2955:
1992:-centered colouring of planar graphs.
1910:are universal for outerplanar graphs.
1087:This result provides an easy proof of
1453:
1187:{\displaystyle D={\frac {f-1}{2v-5}}}
399:Instead of considering subdivisions,
231:
168:as well, and vice versa, by means of
3146:3 Utilities Puzzle and Planar Graphs
3095:
1876:states that every planar graph is 4-
1518:-outerplanar embedding. A graph is
506:for this problem: for a graph with
452:contains a minor isomorphic to the
24:
2020:
1271:corresponding to the two faces in
1119:of bounded faces (the same as the
968:Euler's formula is also valid for
692:
461:graph, and is therefore non-planar
237:Kuratowski's and Wagner's theorems
25:
3188:
3117:
3081:; Sreshta, S. (October 1, 2003).
3074:. Special Issue on Graph Drawing.
2405:Geometric graphs and arrangements
2222:Morphogenesis of Spatial Networks
2118:Colin de Verdière graph invariant
1127:) by its maximal possible values
1004:
596:In this sense, planar graphs are
497:
2742:; Joret, Gwenäel; Micek, Piotr;
1867:
1605:Word-representable planar graphs
202:. Although a plane graph has an
109:
91:
72:
61:
3027:A useful planar graph generator
2893:
2847:
2807:
2779:
2732:
2687:
2646:
2593:
1982:non-repetitive chromatic number
1529:
1303:has faces and as many faces as
1267:connecting the two vertices in
175:Plane graphs can be encoded by
31:. For triangulated graphs, see
3177:Intersection classes of graphs
2436:
2396:
2361:
2326:
2281:
2238:
2209:
2172:
1434:are the graphs in which every
1391:If a maximal planar graph has
1125:Mac Lane's planarity criterion
1038:
642:Mac Lane's planarity criterion
448:An animation showing that the
368:An example of a graph with no
44:
13:
1:
2916:
2632:10.1090/s0894-0347-08-00624-3
1574:A planar graph is said to be
1222:
687:
636:Whitney's planarity criterion
3110:10.1016/0095-8956(78)90059-X
2182:Introduction to Graph Theory
2179:Trudeau, Richard J. (1993).
2092:linklessly embeddable graphs
1622:Enumeration of planar graphs
1446:(without deleting edges) of
395:and is therefore non-planar.
228:" for other related topics.
108:
90:
71:
60:
7:
2413:10.1007/978-3-322-80303-0_1
2247:European Physical Journal B
2123:
2052:simple crossings per edge.
1995:For two planar graphs with
1616:
1450:and maximal planar graphs.
1263:we introduce a new edge in
942:{\displaystyle e\leq 3v-6.}
766:As an illustration, in the
718:is the number of edges and
714:is the number of vertices,
10:
3193:
2681:10.1016/j.jctb.2004.09.007
2267:10.1140/epjb/e2004-00364-9
1888:planar straight-line graph
1042:
696:
26:
3063:10.1142/S0129054106004248
2816:Advances in Combinatorics
2718:10.1007/s00373-006-0647-2
2579:10.1016/j.dam.2016.05.025
2534:10.1007/s00373-016-1693-z
2488:10.1016/j.dam.2015.07.033
2009:graph isomorphism problem
1857:{\displaystyle 30.06^{n}}
1522:-outerplanar if it has a
1368:A simple graph is called
1347:Families of planar graphs
706:states that if a finite,
600:, in that they have only
474:Robertson–Seymour theorem
47:
2901:Graphs and Combinatorics
2696:Graphs and Combinatorics
2166:
2142:Thickness (graph theory)
1929:planar separator theorem
1914:Scheinerman's conjecture
1830:{\displaystyle 27.2^{n}}
1526:-outerplanar embedding.
1402:, then it has precisely
999:topologically equivalent
756:{\displaystyle v-e+f=2.}
335:complete bipartite graph
192:topologically equivalent
170:stereographic projection
3151:NetLogo Planarity model
2948:10.4064/fm-15-1-271-283
2935:Fundamenta Mathematicae
2370:Journal of Graph Theory
2224:. Springer. p. 6.
2216:Barthelemy, M. (2017).
2160:Three utilities problem
2016:class with few cliques.
1943:/3 by the removal of O(
1231:A planar graph and its
667:partial order dimension
2382:10.1002/jgt.3190080206
2071:. More generally, the
1858:
1831:
1804:
1778:
1736:
1710:
1644:
1599:Tutte's spring theorem
1560:directed acyclic graph
1506:a planar embedding is
1365:
1299:, as many vertices as
1235:
1188:
1106:meshedness coefficient
1078:circle packing theorem
1063:
1045:Circle packing theorem
978:perspective projection
965:
943:
818:mathematical induction
757:
462:
396:
313:it does not contain a
282:
3024:; Brinkmann, Gunnar,
2958:Mathematische Annalen
2924:Kuratowski, Kazimierz
2869:10.1145/800141.804671
2746:; Ueckerdt, Torsten;
1859:
1832:
1805:
1779:
1737:
1711:
1645:
1359:
1352:Maximal planar graphs
1291:is intersected. Then
1230:
1189:
1052:
976:of the polyhedron, a
955:
944:
904:in the sense that if
758:
447:
367:
359:• —— • to • — • — • )
244:
3037:Ossona de Mendez, P.
2863:. pp. 236–243.
2096:topologically linked
1841:
1814:
1810:vertices is between
1794:
1746:
1720:
1654:
1634:
1570:Convex planar graphs
1550:Upward planar graphs
1362:Goldner–Harary graph
1146:
1100:Planar graph density
997:that form a surface
918:
885:Euler characteristic
729:
680:Hanani–Tutte theorem
361:zero or more times.
299:Kuratowski's theorem
291:Kazimierz Kuratowski
2624:2009JAMS...22..309G
2259:2004EPJB...42..123B
2218:"1.5 Planar Graphs"
2003:) whether they are
1892:universal point set
1880:(i.e., 4-partite).
1556:upward planar graph
1432:Strangulated graphs
1422:Apollonian networks
1374:plane triangulation
1238:Given an embedding
828:. This lowers both
472:". This is now the
264:and finding either
246:Proof without words
3035:de Fraysseix, H.;
3013:10.7155/jgaa.00091
2970:10.1007/BF01594196
2834:10.19086/aic.27351
2752:Journal of the ACM
2653:McDiarmid, Colin;
2453:10.1007/BFb0071635
2347:10.1007/BF01762117
2304:10.1007/BF00353652
2162:, a popular puzzle
1931:states that every
1918:intersection graph
1874:four color theorem
1854:
1827:
1800:
1774:
1732:
1706:
1640:
1610:Word-representable
1595:3-vertex-connected
1459:Outerplanar graphs
1454:Outerplanar graphs
1366:
1236:
1197:The density obeys
1184:
1080:, first proved by
1064:
983:Steinitz's theorem
966:
939:
753:
663:Schnyder's theorem
463:
397:
283:
232:Planarity criteria
177:combinatorial maps
2987:Myrvold, Wendy J.
2878:978-0-89791-017-0
2786:Bose, Prosenjit;
2758:(4): 22:1–22:38,
2557:Discr. Appl. Math
2512:Graphs and Combin
2475:Discr. Appl. Math
2231:978-3-319-20565-6
2192:978-0-486-67870-2
2130:Combinatorial map
2007:or not (see also
1939:of size at most 2
1803:{\displaystyle n}
1643:{\displaystyle n}
1182:
1135:for a graph with
1123:of the graph, by
987:polyhedral graphs
657:planarity testing
525:planarity testing
262:Wagner's theorems
188:equivalence class
127:
126:
16:(Redirected from
3184:
3112:
3092:
3073:
3056:
3047:(5): 1017–1029,
3030:
3016:
3015:
2995:
2985:Boyer, John M.;
2980:
2951:
2950:
2932:
2911:
2897:
2891:
2890:
2862:
2851:
2845:
2844:
2827:
2811:
2805:
2804:
2803:
2783:
2777:
2776:
2767:
2736:
2730:
2729:
2711:
2691:
2685:
2684:
2674:
2655:Steger, Angelika
2650:
2644:
2643:
2617:
2597:
2591:
2590:
2572:
2552:
2546:
2545:
2527:
2507:
2501:
2500:
2490:
2472:
2463:
2457:
2455:
2440:
2434:
2433:
2400:
2394:
2392:
2365:
2359:
2357:
2341:(1–4): 247–278,
2330:
2324:
2322:
2285:
2279:
2277:
2242:
2236:
2235:
2213:
2207:
2206:
2201:
2199:
2176:
1986:universal graphs
1968:
1967:
1951:
1950:
1908:regular polygons
1863:
1861:
1860:
1855:
1853:
1852:
1836:
1834:
1833:
1828:
1826:
1825:
1809:
1807:
1806:
1801:
1783:
1781:
1780:
1775:
1773:
1772:
1741:
1739:
1738:
1733:
1715:
1713:
1712:
1707:
1696:
1695:
1683:
1682:
1678:
1649:
1647:
1646:
1641:
1588:
1540:polyhedral graph
1525:
1521:
1517:
1509:
1505:
1495:
1491:
1487:
1478:
1469:
1436:peripheral cycle
1417:
1409:
1401:
1394:
1328:
1324:
1316:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1241:
1218:
1211:
1204:
1193:
1191:
1190:
1185:
1183:
1181:
1167:
1156:
1138:
1134:
1118:
1111:
1061:
1034:
1019:
974:Schlegel diagram
970:convex polyhedra
958:Schlegel diagram
948:
946:
945:
940:
910:
882:
867:
860:
849:
836:by one, leaving
835:
831:
815:
808:
794:
790:
783:
776:
762:
760:
759:
754:
721:
717:
713:
630:
621:
610:
591:
577:
563:
548:
541:
537:
533:
522:
509:
493:
484:
470:forbidden minors
460:
428:
419:
401:Wagner's theorem
394:
385:
376:
360:
344:
332:
295:forbidden graphs
181:rotation systems
155:planar embedding
113:
95:
76:
65:
45:
21:
3192:
3191:
3187:
3186:
3185:
3183:
3182:
3181:
3157:
3156:
3120:
2993:
2930:
2919:
2914:
2898:
2894:
2879:
2860:
2852:
2848:
2812:
2808:
2784:
2780:
2774:10.1145/3385731
2737:
2733:
2709:10.1.1.106.7456
2692:
2688:
2651:
2647:
2598:
2594:
2553:
2549:
2508:
2504:
2470:
2464:
2460:
2441:
2437:
2423:
2401:
2397:
2366:
2362:
2331:
2327:
2286:
2282:
2243:
2239:
2232:
2214:
2210:
2197:
2195:
2193:
2177:
2173:
2169:
2126:
2114:Petersen family
2111:
2104:
2023:
2021:Generalizations
1963:
1961:
1946:
1944:
1900:integer lattice
1870:
1848:
1844:
1842:
1839:
1838:
1821:
1817:
1815:
1812:
1811:
1795:
1792:
1791:
1765:
1761:
1747:
1744:
1743:
1721:
1718:
1717:
1691:
1687:
1674:
1667:
1663:
1655:
1652:
1651:
1635:
1632:
1631:
1624:
1619:
1607:
1587:
1583:
1580:convex polygons
1572:
1552:
1532:
1523:
1519:
1511:
1507:
1500:
1493:
1489:
1486:
1480:
1477:
1471:
1468:
1462:
1456:
1448:complete graphs
1411:
1403:
1396:
1392:
1354:
1349:
1326:
1322:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1239:
1225:
1213:
1206:
1198:
1168:
1157:
1155:
1147:
1144:
1143:
1136:
1128:
1113:
1109:
1102:
1060:
1054:
1047:
1041:
1021:
1010:
1007:
995:simple polygons
919:
916:
915:
905:
869:
862:
851:
837:
833:
829:
810:
796:
792:
785:
778:
771:
768:butterfly graph
730:
727:
726:
719:
715:
711:
704:Euler's formula
701:
695:
693:Euler's formula
690:
629:
623:
612:
601:
582:
568:
554:
543:
539:
535:
531:
511:
507:
500:
492:
486:
483:
477:
459:
453:
427:
421:
418:
412:
393:
387:
384:
378:
375:
369:
358:
343:
337:
331:
325:
297:, now known as
277:
270:
250:hypercube graph
239:
234:
226:graph embedding
123:
114:
106:
100:
96:
86:
77:
68:Butterfly graph
66:
48:Example graphs
43:
40:
23:
22:
15:
12:
11:
5:
3190:
3180:
3179:
3174:
3172:Graph families
3169:
3155:
3154:
3148:
3143:
3137:
3131:
3119:
3118:External links
3116:
3115:
3114:
3093:
3091:on 2016-03-16.
3075:
3032:
3022:McKay, Brendan
3018:
3006:(3): 241–273,
2982:
2953:
2918:
2915:
2913:
2912:
2907:(3), 337–352.
2892:
2877:
2846:
2806:
2788:Dujmović, Vida
2778:
2748:Wood, David R.
2740:Dujmović, Vida
2731:
2702:(2): 185–202.
2686:
2672:10.1.1.572.857
2665:(2): 187–205.
2645:
2608:(2): 309–329.
2592:
2547:
2518:(5): 1749–61.
2502:
2458:
2435:
2421:
2395:
2376:(2): 241–251,
2360:
2325:
2298:(4): 323–343,
2280:
2253:(1): 123–129,
2237:
2230:
2208:
2191:
2170:
2168:
2165:
2164:
2163:
2157:
2154:Sprouts (game)
2151:
2145:
2139:
2133:
2125:
2122:
2109:
2102:
2065:toroidal graph
2042:1-planar graph
2022:
2019:
1924:in the plane.
1884:Fáry's theorem
1869:
1866:
1851:
1847:
1824:
1820:
1799:
1771:
1768:
1764:
1760:
1757:
1754:
1751:
1731:
1728:
1725:
1705:
1702:
1699:
1694:
1690:
1686:
1681:
1677:
1673:
1670:
1666:
1662:
1659:
1639:
1623:
1620:
1618:
1615:
1606:
1603:
1597:planar graph.
1585:
1571:
1568:
1551:
1548:
1531:
1528:
1484:
1475:
1466:
1455:
1452:
1440:chordal graphs
1395:vertices with
1386:chordal graphs
1382:complete graph
1370:maximal planar
1353:
1350:
1348:
1345:
1343:) embeddings.
1224:
1221:
1195:
1194:
1180:
1177:
1174:
1171:
1166:
1163:
1160:
1154:
1151:
1101:
1098:
1089:Fáry's theorem
1058:
1043:Main article:
1040:
1037:
1006:
1005:Average degree
1003:
985:says that the
950:
949:
938:
935:
932:
929:
926:
923:
764:
763:
752:
749:
746:
743:
740:
737:
734:
697:Main article:
694:
691:
689:
686:
685:
684:
683:modulo 2.
676:
670:
660:
649:
639:
627:
594:
593:
579:
565:
499:
498:Other criteria
496:
490:
481:
457:
450:Petersen graph
435:
434:
425:
416:
391:
382:
373:
351:
350:
341:
329:
323:complete graph
311:if and only if
289:mathematician
275:
268:
238:
235:
233:
230:
198:, is called a
125:
124:
121:
107:
104:
98:Complete graph
88:
87:
84:
79:Complete graph
70:
58:
57:
54:
50:
49:
41:
9:
6:
4:
3:
2:
3189:
3178:
3175:
3173:
3170:
3168:
3167:Planar graphs
3165:
3164:
3162:
3152:
3149:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3125:
3122:
3121:
3111:
3107:
3103:
3099:
3094:
3090:
3086:
3085:
3080:
3076:
3072:
3068:
3064:
3060:
3055:
3050:
3046:
3042:
3038:
3033:
3029:
3028:
3023:
3019:
3014:
3009:
3005:
3001:
3000:
2992:
2988:
2983:
2979:
2975:
2971:
2967:
2963:
2960:(in German),
2959:
2954:
2949:
2944:
2940:
2937:(in French),
2936:
2929:
2925:
2921:
2920:
2910:
2906:
2902:
2896:
2888:
2884:
2880:
2874:
2870:
2866:
2859:
2858:
2850:
2843:
2839:
2835:
2831:
2826:
2821:
2817:
2810:
2802:
2797:
2793:
2789:
2782:
2775:
2771:
2766:
2761:
2757:
2753:
2749:
2745:
2741:
2735:
2727:
2723:
2719:
2715:
2710:
2705:
2701:
2697:
2690:
2682:
2678:
2673:
2668:
2664:
2660:
2656:
2649:
2641:
2637:
2633:
2629:
2625:
2621:
2616:
2611:
2607:
2603:
2596:
2588:
2584:
2580:
2576:
2571:
2566:
2562:
2558:
2551:
2543:
2539:
2535:
2531:
2526:
2521:
2517:
2513:
2506:
2498:
2494:
2489:
2484:
2480:
2476:
2469:
2462:
2454:
2450:
2446:
2439:
2432:
2428:
2424:
2422:3-528-06972-4
2418:
2414:
2410:
2406:
2399:
2391:
2387:
2383:
2379:
2375:
2371:
2364:
2356:
2352:
2348:
2344:
2340:
2336:
2329:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2292:
2284:
2276:
2272:
2268:
2264:
2260:
2256:
2252:
2248:
2241:
2233:
2227:
2223:
2219:
2212:
2205:
2194:
2188:
2184:
2183:
2175:
2171:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2136:Planarization
2134:
2131:
2128:
2127:
2121:
2119:
2115:
2108:
2101:
2097:
2093:
2088:
2083:
2082:circuit board
2077:
2074:
2070:
2066:
2061:
2058:
2053:
2051:
2047:
2043:
2038:
2036:
2032:
2028:
2018:
2017:
2012:
2010:
2006:
2002:
1998:
1993:
1991:
1987:
1983:
1979:
1975:
1974:graph product
1970:
1966:
1959:
1955:
1949:
1942:
1938:
1934:
1930:
1925:
1923:
1922:line segments
1919:
1915:
1911:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1879:
1875:
1868:Other results
1865:
1849:
1845:
1822:
1818:
1797:
1788:
1785:
1769:
1766:
1762:
1758:
1755:
1752:
1749:
1729:
1726:
1723:
1703:
1700:
1697:
1692:
1688:
1684:
1679:
1675:
1671:
1668:
1664:
1660:
1657:
1637:
1629:
1614:
1611:
1602:
1600:
1596:
1592:
1581:
1577:
1567:
1565:
1561:
1557:
1547:
1545:
1541:
1537:
1527:
1515:
1503:
1497:
1483:
1474:
1465:
1460:
1451:
1449:
1445:
1441:
1437:
1433:
1429:
1427:
1423:
1419:
1415:
1407:
1399:
1389:
1387:
1383:
1379:
1375:
1371:
1363:
1358:
1344:
1342:
1338:
1333:
1330:
1320:
1315:
1311:
1275:that meet at
1247:
1246:
1234:
1229:
1220:
1216:
1209:
1202:
1178:
1175:
1172:
1169:
1164:
1161:
1158:
1152:
1149:
1142:
1141:
1140:
1132:
1126:
1122:
1116:
1107:
1097:
1094:
1093:line segments
1090:
1085:
1083:
1079:
1075:
1074:
1069:
1057:
1051:
1046:
1036:
1032:
1028:
1024:
1018:
1014:
1002:
1000:
996:
992:
988:
984:
979:
975:
971:
963:
960:of a regular
959:
954:
936:
933:
930:
927:
924:
921:
914:
913:
912:
908:
903:
899:
898:
893:
890:In a finite,
888:
886:
883:, i. e., the
880:
876:
872:
865:
858:
854:
848:
844:
840:
827:
823:
819:
813:
807:
803:
799:
788:
781:
774:
770:given above,
769:
750:
747:
744:
741:
738:
735:
732:
725:
724:
723:
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664:
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654:
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647:
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633:
632:
626:
619:
615:
608:
604:
599:
598:sparse graphs
589:
585:
580:
575:
571:
566:
561:
557:
552:
551:
550:
546:
534:vertices and
528:
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366:
362:
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347:utility graph
340:
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132:
120:
117:
116:Utility graph
112:
103:
99:
94:
89:
83:
80:
75:
69:
64:
59:
55:
52:
51:
46:
38:
34:
33:Chordal graph
30:
19:
18:Planar graphs
3101:
3097:
3089:the original
3083:
3054:math/0610935
3044:
3040:
3026:
3003:
2997:
2961:
2957:
2938:
2934:
2904:
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2895:
2856:
2849:
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2809:
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2781:
2755:
2751:
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2699:
2695:
2689:
2662:
2658:
2648:
2615:math/0501269
2605:
2601:
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2560:
2556:
2550:
2515:
2511:
2505:
2478:
2474:
2461:
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2438:
2404:
2398:
2373:
2369:
2363:
2338:
2335:Algorithmica
2334:
2328:
2295:
2289:
2283:
2250:
2246:
2240:
2221:
2211:
2203:
2196:. Retrieved
2181:
2174:
2106:
2099:
2078:
2062:
2054:
2049:
2045:
2039:
2034:
2030:
2024:
2013:
2000:
1996:
1994:
1989:
1978:queue number
1971:
1964:
1958:branch-width
1947:
1940:
1932:
1926:
1912:
1903:
1895:
1882:
1871:
1789:
1786:
1650:vertices is
1625:
1608:
1575:
1573:
1553:
1533:
1530:Halin graphs
1513:
1501:
1498:
1481:
1472:
1463:
1457:
1430:
1420:
1413:
1405:
1397:
1390:
1373:
1369:
1367:
1341:homeomorphic
1334:
1331:
1313:
1309:
1243:
1237:
1214:
1207:
1200:
1196:
1130:
1121:circuit rank
1114:
1103:
1086:
1071:
1067:
1065:
1055:
1030:
1026:
1022:
1016:
1012:
1008:
967:
962:dodecahedron
906:
901:
895:
889:
878:
874:
870:
863:
856:
852:
846:
842:
838:
811:
805:
801:
797:
786:
779:
772:
765:
703:
702:
646:cycle spaces
624:
622:. The graph
617:
613:
606:
602:
595:
587:
583:
573:
569:
559:
555:
544:
529:
518:
513:
501:
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478:
466:Klaus Wagner
464:
454:
436:
422:
413:
398:
388:
379:
370:
352:
338:
326:
307:finite graph
284:
272:
265:
258:Kuratowski's
253:
215:
207:
203:
199:
185:
174:
163:
154:
150:
141:that can be
135:planar graph
134:
131:graph theory
128:
118:
101:
81:
37:Ternary plot
3079:Bader, D.A.
2964:: 570–590,
2941:: 271–283,
2481:: 164–171.
1591:subdivision
1564:NP-complete
1536:Halin graph
1444:clique-sums
1384:and to the
1337:isomorphism
1108:or density
1039:Coin graphs
991:3-connected
868:, yielding
581:Theorem 3.
553:Theorem 1.
403:deals with
355:subdivision
319:subdivision
159:plane curve
151:plane graph
3161:Categories
3104:(3): 374,
2917:References
2825:1907.04586
2801:2007.06455
2765:1904.04791
2744:Morin, Pat
2570:1507.06749
2525:1503.08002
2037:vertices.
2027:apex graph
2005:isomorphic
1980:, bounded
1628:asymptotic
1410:edges and
1378:line graph
1245:dual graph
1223:Dual graph
1139:vertices:
1082:Paul Koebe
688:Properties
659:algorithm;
538:edges and
504:algorithms
317:that is a
309:is planar
254:non-planar
200:planar map
56:Nonplanar
2978:123534907
2842:195874032
2704:CiteSeerX
2667:CiteSeerX
2563:: 60–70.
2320:122785359
2148:Planarity
2087:soldering
2057:map graph
1954:treewidth
1937:subgraphs
1878:colorable
1767:−
1759:×
1753:≈
1727:≈
1724:γ
1698:⋅
1689:γ
1685:⋅
1669:−
1661:⋅
1544:treewidth
1176:−
1162:−
934:−
925:≤
892:connected
736:−
708:connected
280:subgraphs
278:(bottom)
271:(top) or
208:unbounded
196:isthmuses
3071:40107560
2989:(2005),
2926:(1930),
2887:16345164
2726:22639942
2587:26987743
2542:43817300
2497:26796091
2275:14975826
2198:8 August
2124:See also
1906:-vertex
1730:27.22687
1716:, where
1617:Theorems
1073:osculate
315:subgraph
204:external
143:embedded
2640:3353537
2620:Bibcode
2431:2061507
2390:0742878
2355:2709057
2312:1010382
2255:Bibcode
1962:√
1945:√
1426:3-trees
1418:faces.
1205:, with
333:or the
321:of the
248:that a
153:, or a
145:in the
53:Planar
3069:
2976:
2885:
2875:
2840:
2724:
2706:
2669:
2638:
2585:
2540:
2495:
2429:
2419:
2388:
2353:
2318:
2310:
2273:
2228:
2189:
1984:, and
1576:convex
1504:> 1
1479:or of
1400:> 2
1319:sphere
902:sparse
897:simple
887:is 2.
405:minors
287:Polish
256:using
166:sphere
3067:S2CID
3049:arXiv
2994:(PDF)
2974:S2CID
2931:(PDF)
2883:S2CID
2861:(PDF)
2838:S2CID
2820:arXiv
2796:arXiv
2760:arXiv
2722:S2CID
2636:S2CID
2610:arXiv
2583:S2CID
2565:arXiv
2538:S2CID
2520:arXiv
2493:S2CID
2471:(PDF)
2351:S2CID
2316:S2CID
2291:Order
2271:S2CID
2167:Notes
2073:genus
2069:torus
1846:30.06
1593:of a
1558:is a
1380:of a
1321:. If
1312:** =
826:cycle
816:, by
439:minor
431:minor
429:as a
222:genus
218:genus
147:plane
139:graph
137:is a
2873:ISBN
2417:ISBN
2226:ISBN
2200:2012
2187:ISBN
1956:and
1927:The
1890:. A
1872:The
1837:and
1819:27.2
1756:0.43
1742:and
1626:The
1516:– 1)
1360:The
1233:dual
1199:0 ≤
1104:The
1070:(or
1068:kiss
861:and
832:and
822:tree
784:and
678:The
651:The
485:and
285:The
211:face
133:, a
3106:doi
3059:doi
3008:doi
2966:doi
2962:114
2943:doi
2865:doi
2830:doi
2770:doi
2714:doi
2677:doi
2628:doi
2575:doi
2561:213
2530:doi
2483:doi
2479:201
2449:doi
2409:doi
2378:doi
2343:doi
2300:doi
2263:doi
2110:3,3
2105:or
2025:An
2011:).
1969:).
1920:of
1586:2,4
1554:An
1485:2,3
1416:– 4
1408:– 6
1287:or
1259:in
1217:= 1
1210:= 0
1203:≤ 1
1133:– 5
1117:– 1
1033:= 2
1015:≥ 3
909:≥ 3
881:= 2
866:= 1
859:+ 1
814:= 2
789:= 3
782:= 6
775:= 5
628:3,3
590:– 4
586:≤ 2
576:– 4
572:≤ 2
562:– 6
558:≤ 3
547:≥ 3
527:).
491:3,3
458:3,3
426:3,3
420:or
392:3,3
383:3,3
377:or
342:3,3
276:3,3
260:or
252:is
206:or
190:of
186:An
179:or
129:In
122:3,3
3163::
3102:24
3100:,
3065:,
3057:,
3045:17
3043:,
3002:,
2996:,
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2055:A
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1960:O(
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1029:+
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937:6.
911::
894:,
877:+
873:–
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2107:K
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2100:K
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1965:n
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1941:n
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1823:n
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