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The second graph (the Complete Graph ''K''4) is planar because it can be redrawn without intersecting edges, by moving one of the diagonal edges to wrap around the outside. On the other hand, the two graphs shown below are ''not'' planar: It is not possible to redraw these without edge intersections. In fact, these two are the smallest non-planar graphs, a consequence of the characterization below. A planar graph already drawn in the plane is called a plane graph. The equivalence class of topologicaly equivalent drawings on the sphere is called a planar map. Although a plane graph has an external or '''unbounded''' face, none of the faces of a planar map has a particular status. KURATOWSKI'S AND WAGNER'S THEOREMS The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs, now known as Kuratowski's Theorem : :A Finite Graph is planar If And Only If it does not contain a Subgraph that is an ''expansion'' of ''K''5 (the Complete Graph on five Vertices ) or ''K''3,3 ( Complete Bipartite Graph on six vertices, three of which connect to each of the other three). An ''expansion'' of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) and repeating this zero or more times. Equivalent formulations of this theorem, also known as "Theorem P" include :A finite graph is planar if and only if it does not contain a subgraph that is Homeomorphic to ''K''5 or ''K''3,3. In the Soviet Union , Kuratowski's theorem was known as the Pontryagin -Kuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes. By a long-standing academic tradition, such references are not taken into account in determining priority, so the Russian name of the theorem is not acknowledged internationally. Instead of considering expansions, Wagner's theorem deals with Minors : :A finite graph is planar if and only if it does not have ''K''5 or ''K''3,3 as a Minor . Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of " Forbidden Minor s". This is now the Robertson-Seymour Theorem , proved in a long series of papers. In the Language of this theorem, ''K''5 and ''K''3,3 are the forbidden minors for the class of finite planar graphs. OTHER PLANARITY CRITERIA In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast (''n'') whether the graph is planar or not. For a simple, connected, planar graph with ''n'' vertices and ''e'' edges: : Theorem 1. If ''n'' ≥ 3 then ''e'' ≤ 3''n'' - 6 : Theorem 2. If ''n'' > 3 and there are no cycles of length 3, then ''e'' ≤ 2''n'' - 4 The graph ''K''3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it can not be planar. Note that these theorems are ''if'', not ''if and only if'', 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 must be used. For two planar graphs with ''n'' vertices, it is possible to determine in time O(''n'') whether they are Isomorphic or not.
Euler's formula Euler's formula states that if a finite Connected planar graph is drawn in the plane without any edge intersections, and ''v'' is the number of vertices, ''e'' is the number of edges and ''f'' is the number of Faces (regions bounded by edges, including the outer infinitely large region), then v i.e. the , then remove an edge which completes a Cycle . This lowers both ''e'' and ''f'' by one, leaving ''v'' − ''e'' + ''f'' constant. Repeat until you arrive at a tree; trees have ''v'' = ''e'' + 1 and ''f'' = 1, yielding ''v'' - ''e'' + ''f'' = 2. In a finite Connected '' Simple '' 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 ''sparse'' in the sense that ''e'' ≤ 3''v'' - 6 if ''v'' ≥ 3. A simple graph is called maximal planar if it is planar but adding any edge would destroy that property. All faces (even the outer one) are then bounded by three edges, explaining the alternative term '''triangular''' for these graphs. If a triangular graph has ''v'' vertices with ''v'' > 2, then it has precisely 3''v''-6 edges and 2''v''-4 faces. Note that Euler's formula is also valid for simple simple planar graphs. Outerplanar graphs A graph is called outerplanar if it has an embedding in the plane such that the vertices lie on a fixed Circle and the edges lie inside the Disk of the circle and don't intersect. Equivalently, there is some face that includes every vertex. Every outerplanar graph is planar, but the converse is not true: the second example graph shown above (''K''4) is planar but not outerplanar. This is the smallest non-outerplanar graph: a theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subgraph that is an expansion of ''K''4 (the full graph on 4 vertices) or of ''K''2,3 (five vertices, 2 of which connected to each of the other three for a total of 6 edges). All finite or Countably Infinite Trees are outerplanar and hence planar. All outerplanar graphs are 3-colorable! OTHER FACTS AND DEFINITIONS Every planar graph without loops is 4-partite, or 4- Colorable ; this is the graph-theoretical formulation of the Four Color Theorem . It can be shown that every simple planar graph admits an embedding in the plane such that all edges are Straight Line segments which don't intersect. Similarly, 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.
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. EXTERNAL LINKS
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