planar graphs prepared and instructed by arie girshson semester b, 2014 june 2014planar graphs1

Planar Graphsprepared and Instructed by

Arie GirshsonSemester B, 2014

June 2014

Planar Graphs - Background

June 2014

Graph G is planar (embeddable in the plane), if it can be drawn in the plane so that its edges intersect only at their ends. Such drawing of a planar graph G is called Planar Embedding of G ().

Planar Embedding

Planar Graph (G)(vertices, edges)

Plane Graph ()(points, lines)

Planar Graphs 3June 2014

A relevant topology in studies of planar graphs deal with Jordan curves – Simple closed curve (continuous non-self-intersecting curve whose origin and terminus coincide). The union of the edges in a cycle of a plane graph constitutes a Jordan curve.

Jordan curve J partitions the plane into two disjoint open sets, interior of J and exterior of J. Clearly J=int J ∩ ext J.Jordan curve theorem states that any line joining a point in int J to a point in ext J must meet J in some point.

int Jext J

The Jordan Curve Theorem

J

June 2014 Planar Graphs 4

Theorem: K5 is nonplanarProof: By contradiction. Let G be a plane K5 graph. Denote the vertices by Suppose Cycle C= is a Jordan curve in the plane.

𝑣1𝑣2

𝑣3

int C

ext C

Point must lie in int C or ext C. Suppose int C.

𝑣4

Edges divide int C into int C1 int C2 & int C3.

int C1

int C2int C3 must lie in one of the four regions

ext C int C1 int C2 or int C3.

Suppose ext C, then since int C it follows from Jordan curve theorem that edge must cross C.

𝑣5

Hence, contradicts the assumption that G is a plane graph.

Planar Graphs 5

Embedding on a Surface

June 2014

A Graph G is said to be embeddable on a surface S, if it can be drawn in S so that its edges intersect only at their ends. Such drawing is called embedding of G on S.

Embedding of K5 on a torus:

P P

R R

P P

Q

Q• Representation of the torus as a

rectangle in which opposite sides are identified.

Planar Graphs 6

F(G) - Set of faces of a plane graph G - Number of faces of a plane graph G

Duality / Dual Graphs

June 2014

A plane graph G partitions the plane into connected regions. The closures of these regions called faces of G.

𝑣4𝑣3

𝑣2

𝑣1𝑒1

𝑒2𝑒3

𝑒4𝑒5

𝑓 1𝑓 2

𝑓 3

Each plane graph has exactly one unbounded face, called the exterior/outer face. - boundary of a face of a plane graph G.If G is connected, then can be regarded as a closed walk in which each cut edge of G in is traversed twice.

When contains no cut edges, it is a cycle of G.

𝑏 ( 𝑓 2)=𝑣1𝑒3𝑣2𝑒4𝑣3𝑒5𝑣4𝑒1𝑣1


𝑣7

𝑣6𝑣5

𝑣8 𝑣4𝑣3

𝑣2

𝑣1𝑒1

𝑒2𝑒3

𝑒4𝑒5𝑒6

𝑒7

𝑒8

𝑒9𝑒10

𝑒11𝑒12

𝑓 1𝑓 2

𝑓 3

𝑓 4

𝑓 5𝑓 6

If is a cut edge in a plane graph, just one face is incident with . Otherwise, there are two faces incident with . - degree of a face . The number of edges with which it is incident (the number of edges in ). Cut edges being counted twice.

𝑏 ( 𝑓 2)=𝑣1𝑒3𝑣2𝑒4𝑣3𝑒5𝑣4𝑒1𝑣1𝑏 ( 𝑓 5 )=𝑣7𝑒10𝑣5𝑒11𝑣8𝑒12𝑣8𝑒11𝑣5𝑒8𝑣6𝑒9𝑣7

Example:

separates from separates from


Suppose that G is a connected plane graph. To subdivide a face of G, is to add a new edge joining two vertices on its boundary, in such a way that apart from its endpoints, lies entirely in the interior of .

The result is a plane graph G+, with additional new face. All faces of G are also faces of G+, except the subdivided face , which became two new faces ().

𝑓 Subdivision𝑓 1

𝑓 2

𝑒


Given a plane graph G, graph G* can be defined as follows:Each face of G has a corresponding vertex in G*.Each edge e of G has a corresponding edge in G*.

𝑒1∗𝑓 5∗

𝑓 2∗

𝑓 3∗𝑓 1∗

𝑓 4∗

𝑒2∗𝑒3∗

𝑒4∗

𝑒5∗

𝑒6∗𝑒7∗

𝑒8∗

𝑒9∗

𝑒2𝑒3𝑒4𝑒5

𝑒6𝑒7

𝑒8𝑒1

𝑒9𝑓 2

𝑓 3

𝑓 4

𝑓 5

𝑓 1

Two vertices & are joined by an edge iff their corresponding faces & are separated by edge .The graph G* is called the dual graph of G (which is also planar).

Dual Graph

Graph G Graph G*


Straight forward Embedding of G* in the plane - Each vertex is placed in the corresponding face . Each edge is drawn across the corresponding edge of G exactly once (we can always draw the dual graph as a plane graph in this way).

Note: if is a loop of G, then is a cut edge of G* and vice-versa.

𝑓 2∗

𝑓 3∗𝑓 4∗

𝑓 5∗

𝑓 1∗

𝑒1∗𝑒2∗

𝑒3∗

𝑒4∗

𝑒5∗𝑒6∗

𝑒7∗

𝑒9∗

𝑒8∗

𝑓 2𝑓 3

𝑓 4

𝑓 5

𝑓 1


When G is a connected graph, the dual graph G** G (G** is a dual graph of G*), easily seen from the previous figure.

Conclusion: Isomorphic plane graphs may have non-isomorphic duals.

Example: The following plane graphs are isomorphic. Are the dual graphs isomorphic ? NO

The plane graph on the left has a face of degree five, whereas the right one has a face of degree four.


A few relations, which are direct consequences of the definition of G*:

for all .

Theorem: If G is a plane graph, then

Proof: Let G* be the dual of G.Then:


Proposition: Let G be a connected plane graph, and let be an edge of G that is not a cut edge. Then

(G \ )*G* \ *

Proof: is not a cut-edge, therefore the two faces incident with are distinct (denoted by ), become single face (denoted by ). Any face adjacent to or , becomes adjacent to in G \ . All the other faces are not affected by deletion of . In the dual (G \ )*, the corresponding vertices become , while all the other vertices of G* are vertices of (G \ )*.

Furthermore, any vertex of G* that is adjacent to or is adjacent in (G \ )* to , and adjacencies between vertices of (G \ )* other than are the same as in G*.


Dually, we can look at the following (Proposition): Let G be a connected plane graph, and let be a link of G. Then:

(G \ )*G* \ *

𝑓 1𝑓 2

𝑒𝑓 1∗

𝑓 2∗𝑒∗

G and G* G\e and G*\e*

𝑓𝑓 ∗

Proof: Because G is connected, G**G (already seen). Also, because is not a loop of G, the edge * is not a cut edge of G*, so G* \ * is connected. By previous proposition, (G* \ *)* G** \ ** G \ . Follows on taking duals.

Therefore: (G \ )*G* \ *

Planar Graphs 15

Directed Dual Graph

June 2014

Let D be a plane digraph, with underlying plane graph G. Consider a plane dual G* of G. Each arc of G separates two faces of G. As is traversed from its tail to its head, one of these faces lies to the left () of and one to its right (). Same as in not directed graphs, if is a cut-edge, . For each arc of D, we orient the edge of G* that crosses it as an arc by designating the end lying in as the tail of and the end lying in as the head of . The plane digraph D* is the directed plane dual of D.

Connectivity 16June 2014

Example: Directed dual graph.

Planar Graphs 17

Euler’s Formula

June 2014

Simple formula relating the number of vertices, edges & faces in a connected plane graph (established by Euler).Theorem: If G is a connected plane graph, then:

Proof: By induction on (number of faces of G).If , each edge of G is a cut edge (connected graph) spanning a tree. In this case , theorem holds.

∅=1

Suppose that it is true for all connected plane graphs with fewer than faces, and let G be a connected plane graph with faces. Choose an edge of G, that is not a cut edge (connected graph).

Planar Graphs 18

By induction:GGG

June 2014

Then G is a connected plane graph with faces, since the two faces of G separated by combine to form one face of G. 𝑓 1

𝑓 2𝑓 3𝑒

Using the relations (of G):GGG

We obtain:GGG

The theorem follows by the principle of induction.


Corollary: All planar embeddings of a connected planar graph have the same number of faces.Proof: Let G and H be two planar embeddings of a given connected planar graph. Since G H, and , applying Euler’s Formula:

Corollary: If G is a simple planar graph with then Proof: Let G be a simple connected graph with . Then for all , and .

𝑓 1𝑓 2

Recall theorem: , we have . Euler’s theorem implies or .


Corollary: If G is a simple planar graph, then .

Proof: The corollary is trivial for .If , recall: (Book Theorem 1.1)

(From previous corollary)

Follows from the above:*

* Sum of vertices degree is always bigger or equal to minimum vertex degree multiplied by number of vertices.

planar graphs prepared and instructed by arie girshson semester b, 2014 june 2014planar graphs1

Documents