Everything about Planar Graphs totally explained
In
graph theory, a
planar graph is a
graph which can be
embedded in the plane, for example, it can be drawn on the plane in such a way that its edges may intersect only at their endpoints.
A nonplanar graph is the one which can't be drawn in the plane without edge intersections.
A planar graph already drawn in the plane without edge intersections is called a
plane graph or
planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point in 2D space, and from every edge to a
plane curve, 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.
It is easily seen that a graph that can be drawn on the plane can be drawn on the
sphere as well, and vice versa.
The
equivalence class of
topologically 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 have a particular status.
A generalization of planar graphs are graphs which can be drawn on a surface of a given
genus. In this terminology, planar graphs have
graph genus 0, since the plane (and the sphere) are surfaces of genus 0. See "
graph embedding" for other related topics.
Kuratowski's and Wagner's theorems
The
Polish mathematician
Kazimierz Kuratowski provided a characterization of planar graphs in terms of
forbidden graphs, now known as
Kuratowski's theorem:
» A
finite graph is planar
if and only if it doesn't contain a
subgraph that's a
subdivision of
K5 (the
complete graph on five
vertices) or
K3,3 (
complete bipartite graph on six vertices, three of which connect to each of the other three).
A
subdivision 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 doesn't contain a subgraph that's
homeomorphic to
K5 or
K3,3.
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Pontryagin-Kuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes.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 isn't acknowledged internationally.
Instead of considering subdivisions,
Wagner's theorem deals with
minors:
» A finite graph is planar if and only if it doesn't have
K5 or
K3,3 as a
minor.
Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "
forbidden minors". This is now the
Robertson-Seymour theorem, proved in a long series of papers. In the
language of this theorem,
K5 and
K3,3 are the forbidden children for the class of finite planar graphs.
Other planarity criteria
In practice, it's difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast
algorithms for this problem: for a graph with
n vertices, it's possible to determine in time
O(
n) (linear time) whether the graph may be planar or not (see
planarity testing).
For a simple, connected, planar graph with
v vertices and
e edges, the following simple planarity criteria hold:
» Theorem 1. If
v ≥ 3 then
e ≤ 3
v - 6;
Theorem 2. If
v > 3 and there are no cycles of length 3, then
e ≤ 2
v - 4.
In this sense, planar graphs are
sparse graphs, in that they've only O(
v) edges, asymptotically smaller than the maximum O(
v2). The graph
K3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it can't be planar. Note that these theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph isn't planar, not that it's planar. If both theorem 1 and 2 fail, other methods may be used.
For two planar graphs with
v vertices, it's possible to determine in time O(
v) whether they're
isomorphic or not (see also
graph isomorphism problem).
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
»
for example the
Euler characteristic is 2. As an illustration, in the first planar graph given above, we've
v=6,
e=7 and
f=3. If the second graph is redrawn without edge intersections, we get
v=4,
e=6 and
f=4. Euler's formula can be proven as follows: if the graph isn't a
tree, 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's 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
polyhedra. This is no coincidence: every simple polyhedron can be turned into a connected, simple, planar graph by using the polyhedron's vertices as vertices of the graph and the polyhedron's edges as edges of the graph. The faces of the resulting planar graph then correspond to the faces of the polyhedron. For example, the second planar graph shown above corresponds to a
tetrahedron. Not every connected, simple, planar graph belongs to a simple polyhedron in this fashion: the trees do not, for example. A theorem of
Ernst Steinitz says that the planar graphs formed from
convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite
3-connected 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's some face that includes every vertex. Every outerplanar graph is planar, but the converse isn't true: the second example graph shown above (
K4) 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 doesn't contain a subgraph that's an expansion of
K4 (the full graph on 4 vertices) or of
K2,3 (five vertices, 2 of which connected to each of the other three for a total of 6 edges).
Properties of outerplanar graphs
All finite or
countably infinite trees are outerplanar and hence planar.
An outerplanar graph without loops (edges with coinciding endvertices) has a vertex of degree at most 2.
All loopless outerplanar graphs are 3-colorable; this fact features prominently in the simplified proof of
Chvátal's art gallery theorem by . A 3-coloring may be found easily by removing a degree-2 vertex, coloring the remaining graph recursively, and adding back the removed vertex with a color different from its two neighbors.
k-outerplanar graphs
A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For
k > 1 a planar embedding is
k-outerplanar if removing the vertices on the outer face results in a (
k-1)-outerplanar embedding. A graph is
k-outerplanar if it has a
k-outerplanar embedding
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.
Fáry's theorem states 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.
Given an embedding
G of a (not necessarily simple) planar graph in the plane without edge intersections, we construct the
dual graph G* as follows: we choose one vertex in each face of
G (including the outer face) and for each edge
e in
G we introduce a new edge in
G* connecting the two vertices in
G* corresponding to the two faces in
G that meet at
e. Furthermore, this edge is drawn so that it crosses
e exactly once and that no other edge of
G or
G* is intersected. Then
G* is again the embedding of a (not necessarily simple) planar graph; it has as many edges as
G, as many vertices as
G has faces and as many faces as
G has vertices. The term "dual" is justified by the fact that
G** =
G; here the equality is the equivalence of embeddings on the
sphere. If
G is the planar graph corresponding to a convex polyhedron, then
G* is the planar graph corresponding to the dual polyhedron.
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.
A
Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points; see
Geometric graph theory.
Further Information
Get more info on 'Planar Graphs'.
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