planarity: dual graphs - harvey mudd collegekindred/cuc-only/math104/lectures/lect17... ·...

Planarity: dual graphs

Math 104, Graph Theory

March 28, 2013

Math 104, Spring 2013 Planarity: dual graphs

Duality

Definition Given a plane graph G, the dual graph G

⇤ is theplane graph whose vtcs are the faces of G. Thecorrespondence between edges of G and those of G

⇤ is asfollows:

if e 2 E(G) lies on the boundaries of faces X and Y ,

then the endpts of the dual edge e

⇤ 2 E(G⇤) are the

vtcs x and y that represent faces X and Y of G.


Example of a dual graph

A dual graph of a plane graph.


Example of a dual graph

A dual graph of a plane graph.


Correspondence between G and G

⇤

vtcs of G

⇤ =) faces of G

edges of G

⇤ =) edges of G

dual of a plane graph =) multigraph

loops of G

⇤ =) cut edges of G

multiple/paralleledges of G

⇤ =) distinct faces of G w/multiplecommon boundary edges


Observations about a dual graphI The dual G

⇤ of a plane graph G is itself a plane graph.(Intuitively clear that we can draw the dual as a planegraph, but we do not prove this fact.)

I The dual of any plane graph is connected.

Question Given a planar graph G, do all planar embeddings ofG have the same dual (isomorphic duals)?


Observations about a dual graph

I The dual G

⇤ of a plane graph G is itself a plane graph.(Intuitively clear that we can draw the dual as a planegraph, but we do not prove this fact.)

I The dual of any plane graph is connected.

I Two embeddings of a planar graph may havenonisomorphic duals.

I (G⇤)⇤ = G if and only if G is connected.(Requires proof – Exercise 6.1.18 of text.)

Preview: planar graphs and coloring

coloringregions of map !

coloring faces ofplanar embedding ! coloring vtcs of G


Planarity: characterizationsand special classes of planar graphs

Math 104, Graph Theory

March 28, 2013

Math 104, Spring 2013 Planarity

Subdivisions

Definition The subdivision of an edge e = uv is thereplacement of e with new vertex w and two new edges uw andwv (replace edge w/path of length 2).

u

v

u

vw

Definition A graph H is a subdivision of graph G if one canobtain H from G by a series of edge subdivisions.


Example of a subdivision

Petersen graph contains a K3,3-subdivision.


Example of a subdivision

Petersen graph contains a K3,3-subdivision.


Minors

Definition A minor of a graph G is any graph obtainable fromG via a sequence of vertex and edge deletions and edgecontractions.

Petersen graph has a K5-minor.


Subdivisions and minors

LemmaIf G contains a subdivision of H, then H is a minor of G.

As an example, let G be the Petersen graph and H be K3,3. We havealready seen that the Petersen graph contains a subdivision of K3,3.

Furthermore, K3,3 is a minor of the Petersen graph as seen below:

blue = deletered = contract

Petersen graph has a K3,3-minor.


Subdivisions and minors

LemmaIf G contains a subdivision of H, then H is a minor of G.

As an example, let G be the Petersen graph and H be K3,3. We havealready seen that the Petersen graph contains a subdivision of K3,3.

Furthermore, K3,3 is a minor of the Petersen graph as seen below:

blue = deletered = contract

K3,3

Petersen graph has a K3,3-minor.


Characterizations of planar graphs

Theorem (Kuratowski)A graph G is planar if and only if contains no subdivision of K5or K3,3.

Kuratowski’s Theorem...I provides a structural characterization of planar graphsI is useful in showing that a graph is nonplanar

Theorem (Wagner)A graph G is planar if and only neither K5 nor K3,3 is a minor ofG.


Special kinds of planar graphs

Definition A maximal planar graph is a simple planar graphthat is not a spanning subgraph of another planar graph.

In other words, it is a planar graph to which no new edge canbe added without violating the planarity.

Definition A triangulation is a simple plane graph in which allfaces have degree 3.


Maximal planar graphs

PropositionFor a simple n-vertexplane graph G withn � 3, the following areequivalent.

1. G is a maximalplane graph.

2. G is a triangulation.3. G has 3n�6 edges.

Proof.1 ) 2: Suppose BWOC 9 a face ofdegree 4 or more in the maximalplane graph G. Then 9 vtcs u and von this face boundary such thatuv 62 E(G); if not, a vertex andincident edges can be added to theinterior of the face to obtain a planarembedding of K5.

Drawing the edge uv in the interiorof the face does not violate theplanarity of G and hence contradictsthe maximality of G. )(

⌅






Proof.1 ) 2: Suppose BWOC 9 a face ofdegree 4 or more in the maximalplane graph G. Then 9 vtcs u and von this face boundary such thatuv 62 E(G); if not, a vertex andincident edges can be added to theinterior of the face to obtain a planarembedding of K5.

Drawing the edge uv in the interiorof the face does not violate theplanarity of G and hence contradictsthe maximality of G. )( ⌅






Proof.2 ) 3: Suppose G is a triangulationwith f faces. Then

2|E(G)|= Âfaces

d(face) = 3f

and so by Euler’s formula,

2 = n�e+ f = n�e+ 23e

= n� 13e.

Thus, e = 3n�6.⌅






Proof.3 ) 1: By previous theorem,

|E(G)| 3n�6,

so if G has 3n�6 edges, no edgecan be added to the planar graph G.

⌅



Remark Every simple planar graph is a spanning subgraph ofa maximal planar graph.

Procedure to get maximal planar graph from simple plane graph

I In a plane graph G with at least 3 vtcs, if a face has a cutedge inside it, add an edge between one end of cut edgeand any other vertex on boundary of face.

I Otherwise, a face boundary is a k -cycle with k � 3. Ifk � 4, edges (chords) can be drawn from one vertex on thecycle to every other vertex on the cycle so that the face isdivided into triangles.


Example of generating a maximal planar graph

Adding edges to obtain a maximal planar graph.


Example of generating a maximal planar graph

Adding edges to obtain a maximal planar graph.


Outerplanar graphs

Definition A graph is outerplanar if it has a planar embeddingin which all vtcs are on the boundary of the unbounded face.

(Equivalently, some face in the embedding includes everyvertex on its boundary.)

Remarks

I All cycles are outerplanar.

I All trees are outerplanar.

I We can use Kuratowski’s Thm to show that G isouterplanar if and only if G contains no subdivision of K4 orK2,3. (Exercise 6.2.7 of textbook)

I Every simple outerplanar graph has a vertex of degree atmost 2, i.e., d (G) 2.


planarity: dual graphs - harvey mudd collegekindred/cuc-only/math104/lectures/lect17... ·...

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