A STUDY ON DUAL GRAPHS Dissertation submitted to Auxilium College (Autonomous), Vellore – 6 in partial fulfillment of the requirements for the award of the degree of MASTER OF PHILOSOPHY IN MATHEMATICS By Under the Guidance of Postgraduate and Research Department of Mathematics, Auxilium College (Autonomous), Gandhi Nagar, Vellore – 632 006. 1
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A STUDY ON DUAL GRAPHS
Dissertation submitted toAuxilium College (Autonomous), Vellore – 6
in partial fulfillment of the requirementsfor the award of the degree of
MASTER OF PHILOSOPHYIN
MATHEMATICS
By
Under the Guidance of
Postgraduate and Research Department of Mathematics,Auxilium College (Autonomous),Gandhi Nagar, Vellore – 632 006.
July – 2011
1
AUXILIUM COLLEGE (Autonomous)
(Re-Accredited by NAAC with A Grade with a CGPA of 3.41 out of 4)
Gandhi Nagar, Vellore – 632 006
BONAFIDE CERTIFICATE
This is to certify that the dissertation entitled “A STUDY ON DUAL
GRAPHS” submitted by to Auxilium College (Autonomous), Vellore – 6 in partial
fulfillment for the requirement for the award of degree of MASTER OF
PHILOSOPHY in MATHEMATICS is a record of bonafide research work done by
the candidate during the period August 2010 to July 2011 under my guidance and that
the dissertation has not formed the basis for the award of any degree, diploma,
associateship, fellowship on other similar title to any other candidate and the
dissertation represents independent work on the part of the candidate.
……………………………. ……………………………
Head, PG and Research Supervisor and Head, PG and Research
Department of Mathematics, Department of Mathematics,
Auxilium College (Autonomous), Auxilium College (Autonomous),
Gandhi Nagar, Gandhi Nagar,
Vellore – 632006. Vellore – 632006.
Date ……………….. Date………………….
2
DECLARATION
I hereby declare that the M.Phil., dissertation entitled “A STUDY ON DUAL
GRAPHS” has been my original work and that the dissertation has not formed the
basis for the award of any degree, diploma, associateship, fellowship or any other
similar titles.
Place:
Date : Signature of the Student.
3
CONTENTS
1. Introduction
2. Theorems On Dual Graphs
3. Self-dual Graphs
4. A Characterization Of Partially Dual Graphs
5. Applications of Dual Graphs
Conclusion
Bibliography
1
14
24
38
54
4
INTRODUCTION
CHAPTER – I
Section – 1 Introduction to Graph theory 1
Section - 2 Basic Definitions and Examples 5
Section – 3 Theorems on Dual graphs 10
5
THEOREMS ON DUAL GRAPHS
CHAPTER – II
Section – 1 Theorems On Plane Duality 14
Section - 2 Theorems On Combinatorial Dual 17
Section – 3 Some More Theorems On Duality 20
6
SELF-DUAL GRAPHS
CHAPTER – III
Section – 1 Forms Of Self-Duality 24
Section - 2 A Comparison Of Forms Of Self-
Duality
30
Section – 3 Self-Dual Graphs and Matroids 33
7
A CHARACTERIZATION OF PARTIALLY DUAL
GRAPHS
CHAPTER – IV
Section – 1 Ribbon Graphs 38
Section - 2 Partial Duality 42
Section – 3 Partial Duality For Graphs 48
8
APPLICATIONS OF DUAL GRAPHS
CHAPTER –V
Section – 1 Graph Representations 54
Section - 2 Design Through Duality Relation 56
Section – 3 An Application Of Graph Theory in GSM
Mobile Phone Networks
61
9
CHAPTER-1
INTRODUCTION
SECTION-1
INTRODUCTION TO GRAPH THEORY
Why study Graph theory?
Graph theory provides useful set of techniques for solving real-world
problems- particularly for different kinds of optimization.
Graph theory is useful for analyzing “things that are connected to other
things”, which applies at most everywhere.
Some difficult problems become easy when represented using a graph.
There are lots of unsolved questions in Graph theory: Solve one and become
rich and famous.
“Graph Theory” is an important branch of Mathematics, (Euler 1707-1782) is
known as the father of Graph Theory as well as Topology. Graph theory came into existence
during the first half of the 18th century. Graph theory did not start to develop into an
organized branch of Mathematics until the second half of the 19 th century and, there was not
even a book on the subject until the first half of the 20 th century. Graph theory has
experienced a tremendous growth, one of the main reason for this phenomena is the
applicability of Graph theory in other disciplines such as Physics, Chemistry, Biology,
Psychology, Sociology and theoretical Computer science.
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In Physics, Graph theory is applied in Continuum Statistical Mechanics and Discrete
Statistical Mechanics. Graph theory models have been used to study polymer chains of hydro-
carbons and Percolation theory.
The blossoming of a new branch of study in the field of Chemistry “Chemical Graph
theory” is yet another proof of the importance and role of Graph theory.
Applications of Graph theory to Biology are mostly in Genetics, Ecology and
Environment. Genetic mapping and Evolutionary Genetics are very important.
Growth of Graph theory is mainly due to its application to discrete optimization
problems and due to the advent of Computers. Graph theory plays an important role in several
areas of Computer science such as switching theory ands logical design, artificial intelligence,
formal languages, computer graphics, operating systems, compiler writing and information
organization and retrieval. Graph theory is also applied in inverse areas such as Social
sciences, linguistic, Physical sciences, communications engineering and other fields. Graph
theory is a delightful play ground for the explanations of proof of techniques in Discrete
Mathematics.
Many branches of Mathematics begin with sets and relations. Graph theory is no
expectation to this, indeed graph are next only to sets. Graph theory studies relation between
elements, part of what makes graph theory interesting is that graphs can be used to model
situations that occur in real world problems. These problems can then be studied with the aid
of graphs.
To see how graphs can be used to represent these different systems or structures,
consider the following example;
Example
Diagrams of molecules of the chemical compounds methane and propane are shown
below. These can be represented by graphs using points, called vertices, as the atoms of
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carbons and hydrogen present and lines, called edges, as the bonds. Thus, a molecule of
methane is represented by a graph with five vertices and four edges while propane is
represented by a graph with eleven vertices and ten edges.
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Methane
Propane
Graph theory started with Euler who was asked to find a nice path across the seven
Koningsberg bridges.
13
Another early bird was Sir William Rowan Hamilton (1805-1865).
In 1859 he developed a toy based on finding a path visiting all cities in a graph
exactly once and sold it to a toy maker in Dublin. It never was a big success.
14
The (Eulerian) path
should cross over each
of the seven bridges
exactly once
SECTION-2
BASIC DEFINITIONS AND EXAMPLES
GRAPH
A Graph G=(V, E) consists of a pair of V and E. The elements of V are called vertices
and the elements of E are called edges. Each edge has a set of one or two vertices associated
to it, which are called its end points.
DIGRAPH
Let E be an unordered set of two elements subsets of V. If we consider ordered pair of
elements of V then the graph G (V, E) is called a directed graph or digraph.
CYCLE OR CIRCUIT
A Cycle is a closed walk in which all the vertices are distinct except u = v, that is the
initial and terminal points of the walk coincide.
Example
Figure:1
ACYCLIC OR FOREST
A graph G is called acyclic if, it has no cycles.
TREE
A tree is an acyclic connected graph.
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Example
Figure:2
BIPARTITE GRAPH
A Bipartite graph is one whose vertex can be partitioned into two subsets X and Y so
that each edge has one end in X and one end in Y such a partition (X, Y) is called a
Bipartition of the graph.
Example
Figure:3
EDGE CUT
For subsets and of V denote by [ ] the set of edges with one end in and
the other end in . An edge cut of G is a E of the form [ ] where is a non-empty
proper subset of V and =V\S.
BOND OR CUT-SET
A minimal non-empty edge cut of G is called a Bond.
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Example
Edge cut:{ } { } is a Bond.
Figure:4
CONNECTED
A graph G is said to be connected if between every pair of vertices x and y in G, there
always exists a path in G. Otherwise, G is called disconnected.
LOOP
An edge with identical ends is called a loop.
Example
Figure:5
CUT VERTEX
A vertex v of a graph G is a cut-vertex if the edges set E can be partitioned into two
non-empty subsets and such that G ( ) and G ( ) have just the vertex v in common.
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Example
Figure:6
CUT EDGE
An edge set E of a graph G is a cut edge of G if W(G-e)>W(G).In particular, the
removal of a cut edge from a connected graph makes the graph disconnected.
Example
Figure:7
BLOCK
A connected graph that has no cut vertices is called a Block.
TOUR
A Tour of G is a closed walk of G which includes every edge of G at least once.
EULER TOUR
An Euler Tour of G is a tour which includes each edge of G exactly once.
EULERIAN
A graph G is called Eulerian or Euler if it has an Euler Tour.
Example
Figure:8
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PLANAR GRAPH
A graph G is planar if it can be drawn in the plane in such a way that no two edges
meet except at a vertex with which they both are incident. Any such drawing is a plane
drawing of G.
A graph G is non-planar if no plane drawing of G exists.
Example
Figure:9 Plane drawing of K4
OUTER PLANAR
A Planar graph is an Outer Planar graph if it has an embedding on the plane such that
every vertex of the graph is a vertex belonging to the same (usually exterior) region.
FACES
A plane graph G partitions the rest of the plane into a number of arc-wise connected
open sets. The sets are called the faces of G.
Example
Figure:10
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SECTION-3
DUAL GRAPHS AND EXAMPLES
INTRODUCTION
A map on the plane or the sphere can be viewed as a plane graph in which the faces
are the territories, the vertices are places where boundaries meet and the edges are the porties
of the boundaries that join two vertices from any plane graph we can form a related plane
graph called its “Dual”.
DUAL GRAPHS
Let G be a connected planar graph. Then a dual graph G* is constructed from a plane
drawing of G, as follows.
Draw one vertex in each face of the plane drawing: these are vertices of G*. For each
edge e of a plane drawing, draw a line joining the vertices of G* in faces on either side of e:
these lines are the edges of G*.
REMARK
We always assume that we have been presented with a plane drawing of G.
The procedure is illustrated below.
G G*
Figure: 1
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Also if G is a plane drawing of a connected planar graph, then so its dual G*, and we
can thus construct (G*)*, the dual of G*.
(G*)* G*
Figure: 2
The above diagrams demonstrated that the construction that gives rise to G* from G
can be reversed to give G from G*. It follows that (G*)* is isomorphic to G.
EXAMPLE FOR NON-ISOMORPHIC DUAL GRAPHS
Dual graphs are not unique, in the sense that the same graph can have non-isomorphic
dual graphs because the dual graph depends on a particular plane embedding. In Figure:3, red
graph is not isomorphic to the blue graph G because the upper one has a vertex with
degree 6 (the outer region).
Figure: 3
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PROPERTIES
(1) The dual of a plane graph is planar multi graph- a graph that may have loops and
multiple edges.
(2) If G is a connected graph and if G* is a dual of G then G is a dual of G*.
ON THE UNIQUENESS OF DUAL GRAPHS
(1) Consider the graph and its dual *. Also consider the graph and its dual
*
(see Figure: 4).
(2) Observe that graph and are two different planar representations of a same
graph (say, G).
(3) The graph * contains a vertex of degree of degree 5, and the graph *
contains no
vertex of degree 5. Therefore, * and * are non -isomorphic. So, we have that
but * *.
From (3), we may conclude that two isomorphic planar graphs may have distinct
non- isomorphic duals.
G1 G1*
G2 G2*
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Figure: 4
There are many forms of duality in graph theory.
COMBINATORIAL DUAL GRAPH
Let m(G) be the cycle rank of a graph G, m*(G) be the co-cycle rank, and the relative
complement G-H of a subgraph H of G be defined as that subgraph obtained by deleting the
lines of H. Then a graph G* is a combinatorial dual of G if there is one-to-one
correspondence between their sets of lines such that for any choice Y and Y* of
corresponding subsets of lines,
m*(G-Y) = m*(G) – m(Y*)
where <Y*> is the subgraph of G* with the line set Y*.
Whitney showed that the geometric dual graph and combinatorial dual graph are
equivalent, and so may be called “the” dual graph.
RESULT
A graph is plane if and only if it has a combinatorial dual.
WEAK DUAL
The weak dual of an embedded planar graph is the subgraph of the dual graph whose
vertices correspond to the bounded faces of the primal graph.
SOME RESULTS
A planar graph is outer planar if and only if its weak dual is a forest.
A planar graph is a Halin graph if and only if its weak dual is biconnected and outer
planar.
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CHAPTER – 2
THEOREMS ON DUAL GRAPHS
[12]SECTION-1
THEOREMS ON PLANE DUALITY
PROPOSITION 1
The dual of any plane graph is connected.
PROOF
Let G be a plane graph and G* a plane dual of G. consider any two vertices of G*.
There is a curve in the plane connecting them which avoids all vertices of G. The sequence of
faces and edges of G traversed by this curve corresponds in G* to a walk connecting the two
vertices.
DEFINITION
A simple connected plane graph in which all faces have degree three is called a plane
triangulation or, for a short triangulation.
PROPOSITION 2
A simple connected plane graph is a triangulation if and only if its dual is cubic.
DELETION-CONTRACTION DUALITY
Let G be a planar graph and be a plane embedding of G. For any edge e of G, a
plane embedding of G\e can be obtained by simply deleting the line e from . Thus deletion
of an edge from a planar graph results in a planar graph. Although less obvious, the
contraction of an edge of a planar graph also results in a planar graph. Indeed, given any edge
e of a planar graph G and a planar embedding of G, the line e of can be contracted to a
single point (and the lines incident to its ends redrawn). So, that the resulting plane graph is a
planar embedding of G\e.
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The following two propositions show that the operations of contracting and deleting
edges in plane graphs are related in a natural way under duality.
PROPOSITION 3
Let G be a connected plane graph, and let e be an edge of G that is not a cut edge.
Then (G\e)* G*/e*.
PROOF
Because e is not a cut edge, the two faces of G incident with e are distinct; denote
them by and . Deleting e from G results in a amalgamation of and into a single face
f (see Figure: 1). Any face of G that is adjacent to or is adjacent in G\e to f; all other
faces and adjacencies between them are unaffected by the deletion of e.
Correspondingly, in the dual, the two vertices * and * of G* which correspond to
the faces and of G are now replaced by a single vertex of (G\e)*, which we may denote
by f*, and all other vertices of G* are vertices of (G\e)*. Furthermore, any vertex of G* that
is adjacent to * an * is adjacent in (G\e)* to f*, and adjacencies between vertices of (G\
e)* other than v are the same as in G*. The assertion follows from these observations.
(a) (b)
Figure:1 a) G and , b) G\e and
Dually, we have the following proposition.
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PROPOSTITION 4
Let G be a connected plane graph and let e be a link of G. Then (G/e)* G*\e*.
PROOF
Because, G is connected G** G. Also because e is not a loop of G, the edge e* is not
a cut edge of G*, so G*\e* is connected by proposition:3,
(G*\e*)* G**/e** G/e.
The proposition follows on taking duals.
We now apply Propositions 1 and 2 to show that non separable plane graphs have non
separable duals. This fact turns out to be very useful.
THEOREM 5
The dual of a non separable plane graph is non separable.
PROOF
By induction on the number of edges, Let G be a non separable plane graph. The
theorem is clearly true if G has at most one edge, so we may assume that G has at least two
edges, hence no loops or cut edges. Let e be an edge of G. Then either G\e or G/e is non
separable. If G\e is non separable so is (G\e)* G*/e*, by the induction hypothesis and
proposition 3. And we deduce that G* is non separable. The case where G/e is non separable
can be established by an analogous argument.
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[12]SECTION-2
THEOREMS ON COMBINATORIAL DUAL
PROPOSITION 1
Let G be a 2-connected plane multi graph, and let H be its geometric dual. Then H is a
combinatorial dual of G. Moreover, G is a geometric dual graph (and hence a combinatorial
dual) of H.
PROOF
Since the minimal cuts of G are the minimal separating sets of G,
We now have:
(A) If E E(G) is the edge set of a cycle in G, then E* is cut in H.
(B) If E is the edge set of a forest in G, then H-E* is connected.
Imply that H is a combinatorial dual of G. In particular, H is 2-connected contains at
least three vertices (Otherwise, G is a cycle and the claims are easy to verify). To prove that
G is a geometric dual of H, it sufficies to prove that, for each facial cycle C* in H, has only
one vertex in the face F of H bounded by C*, (clearly, G has no edge inside F). But, if G has
two or more vertices in F, then some two vertices of C* can be joined by a simple arc inside F
having only its ends in common with G H. But, this is impossible by the definition of H.
Whitney [wh33a] proved that combinatorial duals are geometric duals. This gives rise
to another characterization of planar graphs.
THEOREM 2 (Whitney [wh33a])
Let G be a 2-connected multigraph. Then G is a planar if and only if it has a
combinatorial dual. If G* is a combinatorial dual of G, then G has an embedding in the plane
such that G* is isomorphic to the geometric dual of G. In particular, also G*is planar, and G
is a combinatorial dual of G*.
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PROOF
By proposition 1, it sufficies to prove the second part of the theorem. The proof will
be done by induction on the number of edges of G. If G is a cycle, then any two edges of G*
are in a 2-cycle and hence G* has only two vertices. Clearly, G and G* can be represented as
a geometric dual pair.
If G is not a cycle, then G is the union of a 2-connected subgraph and a path P such
that P consists of the two end vertices of P. By the induction hypothesis and by the
proposition, “If G* is a combinatorial dual of G and E E(G) is a set of edges of G such that
G-E has only one component containing edges, then G*/e* is a combinatorial dual of G-
e(minus isolated vertices)”, H=G* /E(P*) is a combinatorial dual of . By the induction
hypothesis, and H can be represented as a geometric dual pair, and is also a
combinatorial dual of H.
If , are two edges of P, then *, * are two edges of G* which belong to a cycle
C* of G*. If C* has length at least 3, then it is easy to find a minimal cut in G* containing e,
but not . But, this is impossible since any cycle in G containing also contains . Hence,
all edges of E (P)* are parallel in G* and join two vertices , say, in G*.
Let be the vertex in H which corresponds to , . The edges in H incident with
form a minimal cut in H. Let C be the corresponding cycle in . As E(C)* separates
from H- in H, C is a simple closed curve separating from H- . In particular, C is facial
in .
Let be the two cycles in CUP containing P such that E ( )* is the minimal cut
consisting of the edges incident with , for i=1,2. Now we draw P inside the face F of
bounded by C and represent inside for i=1,2. This way we obtain a representation of G*
as a geometric dual of G.
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PROPOSITION 3
Let G be a 2-connected multigraph and let G* be its combinatorial dual. Then G* is 3-
connected if and only if G is 3-connected.
PROOF
By Theorem 2, it sufficies to prove that G is 3-connected whenever G* is
3-connected. Suppose that this is not a case if G has a vertex of degree 2, then G* has parallel
edges, a contradiction. So, G has minimum degree at least 3. Then we can write G =
where consists of two vertices, E( ) E( ) = , and each of , contains at
least three vertices.
By Theorem 2, G is planar. Then G has a facial cycle C such that C is path
for i=1,2. Clearly, G/E(C) has two edges which are not in the same block.
By proposition, “If, G* is a combinatorial dual of G and E E (G) is a set of edges of
G such that G-E has only one component containing edges, then G*/E* is a combinatorial
dual of G-E (minus isolated vertices)”, and Theorem 2, G*- E(C)* has two edges which are
not in the same block. As E(C)* is the set of edges incident with a vertex of G*, G* is not
3-connected.
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SECTION-3
SOME MORE THEOREMS ON DUALITY
[9]THEOREM 1
A necessary and sufficient condition for two planar graphs and to be duals of
each other is as follows. There is a one-to-one correspondence between the edges in and
the edges in such that a set of edges in forms a circuit if and only if the corresponding
set in forms a cut-set.
PROOF
Let us consider a plane representation of a planar graph G. Let us also draw
(geometrically) a dual G* of G. Then consider an arbitrary circuit in G. Clearly, will
form some closed simple curve in the plane representation of G- dividing the plane into two
areas (Jordan curve Theorem). Thus the vertices of G* are partitioned into non-empty,
mutually exclusive subsets- one and the other outside.
In other words, the set of edges * in G* corresponding to the set in G is a cut-set
in G*. (No proper subset of * will be a cut-set in G*). Likewise it is apparent that
corresponding to a cut-set S* in G* there is a unique circuit consisting of the corresponding
edge-set S in G such that S is a circuit. This proves the necessity of the theorem.
To prove the sufficiency, let G be a planar graph and let be the graph for which
there is a one-to-one correspondence between the cut-sets of G and circuits of , and vice-
versa. Let G* be a dual graph of G. There is a one-to-one correspondence between the circuits
of and cut-sets of G, and also between the cut-sets of G and circuits of G*. Therefore,
there is one-to-one correspondence between the circuits of and G*, implying that and
G* are 2-isomorphic.
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By a theorem, “All duals of a planar graph G are 2-isomorphic; and every graph 2-
isomorphic to a dual of G is also a dual of G”, must be a dual of G.
[7]THEOREM 2
Edges in a plane graph G form a cycle in G if and only if the corresponding dual
edges form a bond in G*.
PROOF
Consider D E(G). If D contains no cycle in G, then D encloses no region. It remains
possible to reach the unbounded face of G from every face without crossing D. Hence, G*-D*
connected, and D* contains no edge cut.
If D is the edge set of a cycle in G, then the corresponding edge set D* E(G*)
contains all dual edges joining faces inside D to faces outside D. Thus D* contains an edge
cut.
If D contains a cycle and more, then D* contains an edge cut and more.
Thus D* is a minimal edge cut if and only if D is a cycle.
Figure:1
[7]THEOREM 3
The following are equivalent for a plane graph G.
(A) G is bipartite.
(B) Every face of G has even length.
(C) The dual graph G* is Eulerian.
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PROOF
A B. A face boundary consists of closed walks. Every odd closed walk contains an
odd cycle. Therefore, in a bipartite plane graph the contributions to the length of faces are all
even.
B A. Let C be a cycle in G. Since G has no crossings, C is laid out as a simple
closed curve; let F be the region enclosed by C. Every region of G is wholly within F or
wholly outside F. If we sum the face lengths for the regions inside F, we obtain an even
number. Since each face length is even. This sum counts each edge of C once. It also counts
each edge inside F twice, since each such edge belongs twice to faces in F. Hence, the parity
of the length of C is the same as the parity of the full sum, which is even.
B C. The dual graph G* is connected and its vertex degrees are the face lengths of G.
Figure:2
[12]THOREM 4
A graph has a dual if and only if it is planar.
PROOF
We need to prove just the “only if” part. That is, we have only to prove that a non-
planar graph does not have a dual. Let G be a non-planar graph. Then G contains or
or a graph homeomorphic to either of these. We have already seen that a graph G can have a
dual only if every subgraph g of G and every homeomorphic to g has a dual. Thus if we can
32
show that neither nor has a dual, we have proved the theorem. This we shall prove
by contradiction as follows:
(a) Suppose that has a dual D. Observe that the cut-sets in correspond to
circuits in D and vice versa, since has no cut-set consisting of two edges, D has no
circuit consisting of two edges. D contains no pair of parallel edges. Since every circuit in
is of length four or six, D has no cut-set with less than four edges. Therefore, the degree
of every vertex in D is at least four. As D has no parallel edges and the degree of every vertex
is at least four, D must have at least (5 4)/2= 10 edges. This is a contradiction, because
has nine edges and so must its dual. Thus cannot have a dual. Likewise,
(b) Suppose that the graph has a dual H. Note that has (1) 10 edges, (2) no
pair of parallel edges, (3) no cut-set with two edges, and (4) cut-sets with only four or six
edges. Consequently, graph H must have (1) 10 edges, (2) no vertex with degree less than
three, (3) no pair of parallel edges, and (4) circuits of length four and six only. Now graph H
contains a hexagon ( a circuit of length six ), and no more than three edges can be added to a
hexagon without creating a circuit of length three or a pair of parallel edges. Since both of
these are forbidden in H and H has 10 edges, there must be at least seven vertices in at least
three. The degree of each of these vertices is atleast three. This leads to H having at least 11
edges. A contradiction.
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[6]CHAPTER-3
SELF- DUAL GRAPHS
SECTION-1
FORMS OF SELF-DUALITY
DEFINITION
A planar graph is isomorphic to its own dual is called a self-dual graph.
Example
is a Self-dual graph.
Figure: 1
FORMS OF SELF-DUALITY
DEFINITION
Given a planar graph G =(V,E), any regular embedding of the topological realization
of G into a sphere partitions the sphere into regions called the faces of the embedding, and we
write the embedded graph, called a map, as M =(V,E,F). G may have loops and parallel
edges.
DEFINITION
Given a map M, we form the dual map, M* by placing a vertex f* in the centre of
each face f, and for each edge e of M bounding two faces and , we draw a dual edge e*
connecting the vertices * and * and crossing e once transversely. Each vertex v of M
will then correspond to a face v* of M* and we write M* = (F*, E*, V*).
34
If, the graph G has distinguishable embeddings, then G may have more than one dual
graph, see Figure: 2. In this example a portion of the map (V, E, F) is flipped over on a
separating set of two vertices to form (V, E, ).
(V, E, F) * (F*, E*, V*)
(V, E, ) * (F*, E*, V*)
Figure:2
Such a move is called Whitney flip, and the duals of (V, E, F) and (V, E, ) are said
to differ by a Whitney twist. If the graph (V, E) is 3-connected, then there is a unique
embedding in the plane and so the dual is determined by the graph alone.
Given a map X = (V, E, F) and its dual X* = (F*, E*, V*), there are three notions of
self-duality. The strongest, map self-duality, requires that X and X* are isomorphic as maps,
that is, there is an isomorphism : (V, E, F) (F*, E*, V*) preserving incidences. A weaker
notion requires only a graph isomorphism : (V, E) (F*, E*), in which case we say that
the map (V, E, F) is graph self-dual, and we say that G =(V, E) is a self-dual graph.
35
DEFINITION
A geometric duality is a bijection g: E(G) E(G*) such that e E is the edge dual to
g(e) E(G*). If M is 2-cell, then M is connected; so if M is a 2-cell embedding, then
(M*)* M (we use * to indicate the geometric dual operation).
DEFINITION
An algebraic duality is a bijection g: E(G) E( ) such that P is a circuit of G if and
only if g(p) is a minimal edge-cut of . Given a graph G =(V,E), an algebraic dual of G is a
graph for which there exist an algebraic duality g: E(G) E( ).
(a) (b)
(c) (d)
Figure 3: A graph and several of its embeddings.
The geometric duals are shown in dotted lines. Embedding b) is map self-dual, c) is
graphically self-dual and d) is algebraically self-dual.
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We now define several forms of self-duality. Let G =(V, E) be a graph and let
M=(V, E, F) be a fixed map of G, with geometric dual M* =(F*, E*, V*).
DEFINITION
1. M is map self-dual if M M*.
2. M is graphically self-dual if (V, E) (F*, E*).
3. G is algebraically self-dual if G G*, where is some algebraic dual of G.
REMARK
In the literature, the term matroidal or abstract is sometimes used where we use
algebraic.
We will use the geometric duality operation and, unless specified, we will describe a
graph as self-dual if it is graphically self-dual. Since, the dual of a graph is always connected,
we know that a self-dual graph is connected.
The following are a few known results about self-dual graphs.
COROLLARY 1
Let M =(V, E, F) be a 2-cell embedding on an orientable surface. If M is self-dual,
then is even.
PROOF
Since M is self-dual, By Theorem (Euler),
“Let M =(V, E, F) be a 2-cell embedding of a graph in the orientable surface of genus
k. Then, - + = 2-2k”.
= 2-2k- -
= 2(1-k- ).
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THEOREM 2
The complete graph has a self-dual embedding on an orientable surface, if and
only if n 0 or 1 (mod 4).
THEOREM 3
For w 1, there exists a self-dual embedding of some graph G of order n on if
and only if n 4w+1.
Note that a self-dual graph need not be self-dual on the surface of its genus. A single
loop is planar; however it has a (non 2-cell) self-dual embedding on the torus.
Also note that there are infinitely many self-dual graphs. One such infinite family for
the plane is the wheels. A wheel consists of cycle of length n and a single vertex adjacent
to each vertex on the cycle by means of a single edge called a Spoke. The complete graph on
four vertices is also . See Figure: 4 for .
Fig: 4 The 6-Wheel and its dual
MATROIDS
Matroids may be considered a natural generalization of graphs. Thus when discussing
a family of graphs, we should also consider the matroidal implications.
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DEFINITION
Let S be a finite set, the ground set, and let I be a set of subsets of S, the independent
sets. Then = is a matroid if:
1. ;
2. If , then ; and
3. For all A S, all maximal independent subsets of A have the same cardinality.
An isomorphism between two matroids = and = is a bijection
: such that I if and only if (I) . If such a exists, then and are
isomorphic denoted
Given a graph G = (V, E), the cycle matroid (G) of G is the matroid with ground
set E, and F E is independent if and only if F is a forest. A matroid is graphic if there
exists a graph G such that = (G).
For a matroid = (S,I) the dual matroid (S,I*) has ground set S and in
I* if there is a maximal independent set B in such that I S\B. A matroid is
co-graphic if is graphic. It is easily shown that if G is a connected planar graph, then
* (G) = (G*).
It is well known that G is algebraically self-dual if and only if cycle matroids of G and