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Transcript of Maths Mphil Project
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.
10
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
11
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.
12
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.
15
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.
16
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.
17
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
18
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
19
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
20
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
21
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*
22
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.
23
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.
24
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.
26
[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*.
27
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.
28
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.
29
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.
30
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.
31
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.
33
[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.
36
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- ).
37
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.
38
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
G* are isomorphic.
39
SECTION-2
A COMPARISON OF FORMS OF SELF-DUALITY
It is clear that for a map (V, E, F) we have,
Map self-duality Graph self-duality Matroid self-duality.
However, In general, these implications cannot be reversed, as shown by Figure: 3.
But, we are concerned to what extent these implications can be reversed. The next two
theorems assert that, in the most general sense, they cannot.
THEOREM 1
There exist a map (V, E, F) such that (V, E) (E*, V*), but (V, E, F) (F*, E*, V*).
THEOREM 2
There exist a map (V, E, F) such that M(E) M(E*)*, but (V, E) (F*, E*).
SELF-DUAL MAPS AND SELF-DUAL GRAPHS
In the previous examples the graphs were of low connectivity, a planar 3-connected
simple graph has a unique embedding on the sphere, in the sense that if p and q are
embeddings, then there is a homeomorphism h of the sphere so that p =hq. Any isomorphism
between the cycle matroids of a 3-connected graph is carried by a graph isomorphism. Thus,
for a 3-connected graph
Map self-duality Graph self-duality Matroid self-duality,
So self-dual 3-connected graphs, as well as self-dual 3-connected graphic matroids,
reduce to the case of self-dual maps.Since, the examples in Figure:3 are only 1-connected, we
must consider the 2-connected case. In Figure: 5 we see an example of a graphically self-dual
map whose graph is 2-connected which is not map self-dual. One might hope that, as was the
case in Figure:3, that such examples can be corrected by re-embedding or rearranging,
however we have the following strong result.
40
THEOREM 3
There exists a 2-connected map (V, E, F) which is graphically self-dual, so that
(V, E) (F*, V*), but for which every map ( ) such that M(E) M( ) is not map
self-dual.
PROOF
Consider the map in Figure:5 which is drawn on an unfolded cube. The graph is
obtained by gluing two 3-connected self-dual maps together along an edge (a,b) and
Figure: 5.
erasing the common edge. One map has only two reflections as self-dualities, both fixing the
glued edge; the other has only two rotations of order four as dualities, again fixing the glued
edge. The graph self-duality is therefore a combination of both, an order 4 rotation followed
by a Whitney twist of the reflective hemisphere. It is easy to see that all the embeddings of
this graph, as well as the graph obtained after the Whitney flip have the same property.
41
We also have the following.
THEOREM 4
There is a graphically self-dual map (V, E, F) with (V, E) 1-connected and having
only 3-connected blocks, but for which every map ( ) such that M( ) M( ) is not
map self- dual.
PROOF
Consider the 3-connected self-dual maps in Figure: 6. has only self-dualities of
order 4, two rotations and two flip rotations, while has only a left-right reflection and a
rotation as a self-duality. Form a new map X by gluing two copies of to in the
quadrilateral marked with q’s, with the gluing at the vertices marked v and v*. X is
graphically self-dual, as can easily be checked, but no gluing of two copies of can give
map self-duality since every quadrilateral in has order 4 under any self-duality.
Figure: 6
In particular, self-dual graphs of connectivity less than 3 cannot in general be re-
embedded as self-dual maps.
42
SECTION-3
SELF-DUAL GRAPHS AND MATROIDS
If G is 1-connected, then its cycle matroid has a unique decomposition as the direct
sum of connected graphic matroids, (G) = , and if G* is a planar
dual of G, then M(G*) =M(G)* = . If G is a graph self-dual, then
there is a bijection : M(G) M(G*) sending cycles to cycles, and so there is a partition
of {1,2,…….k} such that : , and we that M(G) is the direct sum of self-dual
connected matroids, together with some pairs of terms consisting of a connected matroid and
its dual.
In the next theorem we see that not every self-dual matroid arises from a self-dual
graph.
THEOREM 1
There exists a self-dual graphic matroid M such that for any graph G =(V,E) with
M(G) =M, and any embedding (V, E, F) of G, (V, E) (F*, E*).
PROOF
Consider and , the cycle matroids of two distinct 3-connected self-dual maps
and whose only self-dualities are the antipodal map.
The matroid is self–dual, but its only map realizations are as the
1-vertex union of and , which cannot be self-dual since the cut vertex cannot
simultaneously be sent to both “antipodal” faces.
So for 1-connected graphs, the three notions of self-duality are all distinct. For
2-connected graphs, however we have the following.
43
THEOREM 2
If G =(V, E) is a planar 2-connected graph such that M(E) M(E)*, then G has an
embedding (V, E, F) such that (V, E) (F*, E*).
PROOF
Let (V, E, F) be any embedding of G. Then G is 2-isomorphic, in the sense of
Whitney [15] to (F*, E*), and thus there is a sequence of Whitney flips which transform
(F*, E*, V*) into an isomorphic copy of G and act as re-embeddings of G. Thus the result is a
new embedding of G such that (V, E, F) .
Thus, to describe 2-connected self-dual graphs it is enough up to embedding, to
describe self-dual 2-connected graphic matroid.
SELF-DUAL MATROIDS
DEFINITION
A polyhedron P is said to be self-dual if there is an isomorphism : P P*, where P*
denotes the dual of P. we may regard as a permutation of the elements of P which sends
vertices to faces and vice versa, preserving incidence.
As noted earlier 3-connected self-dual graphic matroids are classified via self-dual
polyhedra.
On the other hand, 1-connected self-dual matroids are easily understood via the direct
sum. Also we show how a 2-connected self-dual matroid M with self-duality arises via
3-connected graphic matroids by recursively constructing its 3-block tree T(M) by adding
orbits of pendant nodes.
The following theorem shows that this construction is sufficient to obtain all
2-connected self-dual matroids.
44
THEOREM 3
Let M be a self-dual connected matroid with 3-block tree T. Let be the tree
obtained from T by deleting all the pendant nodes, and let be the 2-connected matroid
determined by . Then is also self-dual.
PROOF
Let M be a self-dual connected matroid on a set E, so there is a matroid isomorphism
: M M*, so is a permutation of E sending cycles to co-cycles. The 3-block tree of M*
is obtained from that of M by replacing every label with the dual label, so corresponds to a
bijection ( ) of T onto itself, such that for each node of T, : sends
cycles of to co-cycles of . The restriction of ( ) to has the same property
and so corresponds to a self-dual permutation of .
THEOREM 4
Suppose M is a self-dual 2-connected matroid with self-dual permutation and let
. Let be the orbit of under . Suppose one of the following:
(1) k is even and is a 3-connected matroid or a cycle and is a matroid
automorphism of fixing an edge .
(2) k is odd and is a 3-connected self-dual matroid with self-dual permutation
fixing an edge .
For i =1, 2,…., k set and . Let be the matroid obtained
from M by 2-sums with the matroids , amalgamating or * in with .
Let be defined by (e) for e , : is
induced by * for i =1, 2,…., k and . Then is a 2-connected self-dual
matroid with self-dual permutation .
Moreover, every 2-connected self-dual matroid and its self-duality is obtained in this manner.
45
PROOF
The fact that this construction gives a 2-connected self-dual matroid follows at once,
since to check if is a self-duality, it sufficies to check that sends cycles to co-cycles
on each 3-block. The fact that must be self-dual if K is odd follows by considering that
is a self-duality and maps = onto itself.
To see that all self-dualities arise this way, let be a self-duality, let
be a pendant node of T, and set . Let M be the self-dual matroid that results from
removing from the K nodes corresponding to the orbit of the node . induces
M M. Then the desired is .
THE STRUCTURES OF SELF-DUAL-GRAPHS
Given the results of the previous section, we may construct all 2-connected self-dual
graphs; start with any self-dual 2-connected graphic matroid M and chose any realization of
M as a cycle matroid of a graph G. Theorem:2, asserts that G has an embedding as a self-dual
graph. Alternatively, we may carry out a recursive construction in the spirit of Theorem:5 at
the graph level, paying careful attention to the orientations in the 2-sums. The following
theorem gives a more geometric construction.
THEOREM 5
Every 2-connected self-dual graph is 2-isomorphic to a graph which may be
decomposed via 2-sums into self-dual maps such that the 2-sum on any two of the self-dual
maps is along two edges, one of which is the pole of a rotation of order 4 and the other an
edge fixed by a reflection.
PROOF
In case I of Theorem:4, we can always choose to be the identity, and simply glue
in the copies of the maps corresponding to and * compatibly to make a self-dual
map.
46
In case 2 we must have that is a self-dual 3-block containing a self-duality fixing
, hence it corresponds to a self-dual map and must be a reflection or an order 4 rotation
fixing , and likewise the 3-block to which it is attached must be such an edge. If both are of
the same kind, then the 3-blocks may be 2-summed into a self-dual map. This leaves only the
mismatched pair.
Figure: 7
To see that 2-isomorphism is necessary in the above, consider the self-dual graph in
Figure:7. The map cannot be re-embedded as a self-dual map, nor does it have a 2-sum
decomposition described as above, the graph is 2-isomorphic to a self-dual map.
47
ef
[8]CHAPTER-4
A CHARACTERISATION OF PARTIAL DUAL GRAPHS
SECTION-1
RIBBON GRAPHS
S. Chmutov recently introduced the concept of the partial dual of a ribbon graph
G. Partial duality generalizes the natural dual (or Euler- Poincare dual or geometric dual) of a
ribbon graph by forming the dual of G with respect to a subset of its edges A. In contrast with
natural duality, where the topologies of G and G* are similar, the topology of a partial dual
G* can be very different from the topology of G.
For Example,
Although a ribbon graph and its natural dual always have the same genus, a ribbon
graph and a partial dual need not.
THEOREM 1 (EDMONDS CRITERIA)
A 1-1 correspondence between the edges of two connected graphs is a duality with
respect to some polyhedral surface embedding if and only if for each vertex v of each graph,
the edges which meet v correspond in the other graph to the edges of a subgraph which is
Eulerian. That is is connected and has an even number of edge-ends to each of its vertices
(where if an edge meets v both ends its image in is counted twice).
RIBBON GRAPHS
DEFINITION
A ribbon graph G = ( (G), (G)) is (possibly non-orientable) surface with boundary
represent as the union of two sets of topological discs: a set (G) of vertices, and set of edges
(G) such that
48
(1) The vertices and edges intersect in disjoint line segment.
(2) Each such line segment lies on the boundary of precisely one vertex and precisely
one edge;
(3) Every edge contains exactly two such line segments.
It will be convenient to use a description of a ribbon graph G as a spanning sub-
ribbon graph equipped with a set of colored arrows that record where the missing edges.
= =
(i) (ii) (iii)
Figure: 1 Realizations of a ribbon graph.
DEFINITION
An arrow marked ribbon graph consists of a ribbon graph G equipped with a -
collection of colored arrows, called marking arrows, on the boundaries of its vertices. The
marking arrows are such that no marking arrow meets an edge of the ribbon graph, and there
exactly two marking arrows of each other.
ILLUSTRATION
A ribbon graph can be obtained from an arrow-marked ribbon graph by adding edges
in a way prescribed by the marking arrows, thus: take a disc and orient its boundary
arbitrarily. Add this disc to the ribbon graph by choosing two non-interesting arcs on the
boundary of the disc and two marking arrows on the same color, and then identifying the arcs
with the marking arrows according to the orientation of the arrow.
49
The disc that has been added forms an edge of a new ribbon graph.
This process is illustrated in the diagram below, and an example of an arrow-marked
ribbon graph and the ribbon graph it describes in figure 1 (i) and (ii).
Figure: 2
RESULT 2
An arrow-marked ribbon graph describes a ribbon graph. Conversely, every ribbon
graph can be described as an arrow-marked spanning sub-ribbon graph.
PROOF
Suppose that G is a ribbon graph and B (G).
To describe G as an arrow-marked ribbon graph , start by arbitrarily orienting
each edge in B. This induces an orientation on the boundary of each edge in B. To construct
the marking arrows; for each e B, place an arrow on each of the two arcs where e meets
vertices of G, the direction of this arrow should follow the orientation of the boundary e;
color the two arrows with e; and delete the edge e. This gives a marked ribbon graph .
Moreover, the original ribbon graph G can be recovered from by adding edges of
as prescribed by the marking arrows.
Notice that, if G is a ribbon graph and H is any spanning sub-ribbon graph, then there
is an arrow marked ribbon graph of which describes G.
DEFINITION
An arrow presentation of a ribbon graph consists of a set of oriented (topological)
circles (called cycles) that are marked with colored arrows called marking arrows, such that
there are exactly two marking arrows of each color.
50
EXAMPLE
An example of a ribbon graph and its arrow presentation is given in below figure.
=
Figure: 3
Two arrow presentations are considered equivalent if one can be obtained from the
other by reversing pairs of marking arrows of the same color.
51
SECTION-2
PARTIAL-DUALITY
As mentioned above, partial duality is a generalization of the natural dual of a ribbon
graph. A key feature of partial duality is that it provides a way extend the well known relation
T(G; x, y) =T(G*, y, x), relating the Tutte polynomial of a planar graph and its dual, to the
weighted ribbon graph polynomial.
In this section we give a definition of partial duality and then go on to discuss the
relationship between partial duals and naturally dual arrow marked ribbon graphs.
PARTIAL DUALITY
Although the construction of the partial dual of g is perhaps a little lengthy to
write down, in practice the formation of the partial dual is a straightforward process.
DEFINITION
Let G be a ribbon graph and . The partial dual of G along A is defined
below.
(Step P1): Give every edge in (G) orientation (this need not extend to an orientation of the
whole ribbon graph ). Construct a set of marked, oriented, disjoint paths on the boundary of
the edges of G in the following way:
(1) If e A then the intersection of the edge e with distinct vertices (or vertex if e is a
loop) defines two paths. Mark each of these paths with an arrow which points in the
direction of the orientation of the boundary of the edge. Color both of these marks
with e.
(2) If e A then the two sides of e which do not meet the vertices define the two paths.
52
Mark each of these paths with an arrow which points in the direction of the
orientation of the boundary of the edge. Color both of these marks with e.
(Step P2): Construct a set of closed curves on the boundary of G\ by joining the marked
paths constructed above by connecting them along the boundaries of G\ in the natural way.
(Step P3): This defines a collection of non-interesting, closed curves on the boundary of G\
which are marked with colored, oriented arrows. This is precisely an arrow presentation
of a ribbon graph. The corresponding ribbon graph is the partial dual of G*.
The construction is shown locally at an edge e in Figure: 4
An untwisted edge e If e A If e A
A twisted edge e If e A If e A
Figure: 4 Forming paths in the partial dual.
Two examples of the construction of a partial dual are shown below.
EXAMPLE 1
G with A = {2, 3}
53
Steps P1 and P2 Step P3
GA Redrawing GA
Figure: 5
EXAMPLE 2
G with A = {2,3}
Steps P1 and P2 Step P3
GA
54
Figure: 6
Notice that there is a correspondence between the edges G and : every edge of G
gives rise to exactly two marking arrows of the same color, and one edge of is attached
between these two arrows. We will denote the resulting natural bijection between the edge
sets by .
NATURL DUALITY
Before continuing, we will record a few properties of partial duality. We are
particularly interested in the connection between partial and natural duality.
DEFINITION
Let G = ( ) be a ribbon graph. We can regard G as a punctured surface. By
filling in the punctures using a set of discs denoted . We obtain a surface without
boundary . The natural dual (or Euler-Poincare dual) of G is the ribbon graph
G* = ( ).
DUAL EMBEDDING
A dual embedding {G, H, } of G and H into a surface to be an embedding of G
in a surface without boundary which has the property that H = \ (G)
Note that a dual embedding is independent of the order of the ribbon graphs G and H
(i.e. the dual embeddings {G, H, } and {H, G, } are equivalent).
NOTE
The ribbon graphs G and H are natural duals if and only if there exists a dual
embedding {G, H, }.
We can now describe a property of partial duality.
PROPERTY 3
Let G be a ribbon graph, A and \A. Then .
55
PROOF
If , then the cycles defining the vertices of follow the vertices incident with
e in G (See Figure: 2). It then follows that we can delete the edges in before or after
forming the partial dual and end up with the same ribbon graph. Thus
But,
PARTIAL DUAL EMBEDDINGS
DEFINITION
A set {G,H, } is a partial dual embedding of ribbon graph G and H if
i) {G,H, } is a dual embedding;
ii) is a set of disjoint colored arrows marked on the boundaries of the embedded
vertices in with the property that there are exactly two arrows of each color.
THEOREM 4
Let G and H be ribbon graphs. Then G and H are partial duals if and only if there
exists a partial dual embedding { } with the property that is an
arrow-marked ribbon graph describing G, and is an arrow-marked ribbon graph
describing H.
PROOF
First suppose that G and H are partial duals. Then there exists a set of edges A (G)
such that = H. Then G can described as an arrow-marked ribbon graph , where
(A)\A. Let be the surface obtained from by filling in the punctures. Then {
} forms a natural dual embedding. The arrow markings on induce a
56
set of colored arrows on with the property that there are exactly two
arrows of each color. Denote this induced set of colored arrows by . Then
is a partial dual embedding. Moreover,
describes G by construction, and clearly describes =H if we use the
construction of partial duality from the lemma,
“Let G be a ribbon graph, and (A)\A. Then the following
construction gives :
(Step ) : Present G as the arrow-marked ribbon graph .
(Step ): Take the natural dual of The marking arrows on
induce marking arrows on .
(Step ): is the ribbon graph corresponding to the arrow-marked ribbon
graph ”.
Conversely, suppose that { } is a partial dual embedding with the property
that is an arrow-marked ribbon graph describing G, and is an
arrow marked ribbon graph describing H. Then and are precisely the naturally dual
marked ribbon graphs described in step of the construction of partial dual. Here A is the
set of edges of G that are also in .
COROLLARY 5
Let G be a ribbon graph and (G). Then
(1) where ;
(2) (G\A);
(3) when G is orientable, g( )= (2k(G)+e(G)- (G\ )- (G\A)).
57
SECTION-3
PARTIAL DUALITY FOR GRAPHS
DEFINITION
If G=( ) is a ribbon graph then we can construct a graph G =( )
form G by replacing each edge of G with a line, and then contracting the vertices of G into
points, such a graph G is called the core of G.
Notice that there is a natural correspondence between the edges of a ribbon graph and
its core, and the vertices of a ribbon graph and its core.
DEFINITION
We say that two graphs are partial duals if they are cores of partially dual ribbon
graphs.
Let G be a ribbon graph and (G). By the notation we mean that is the
core of where G is the core of G and A is the edge set of G that corresponds with A.
We have seen that partially dual ribbon graphs can be characterized by the existence
of an appropriate partially dual embedding. A corresponding result holds partial dual graphs.
To describe the corresponding result, we make the following definition.
DEFINITION
A partial dual embedding of graphs is a set{ }
Where is a surface without boundary are embedded graphs and E is a set
of colored edges that are embedded in such that
(1) Only the ends of each embedded edge in E meet ;
(2) { } is dual embedding;
(3) Each edge in E is incident to one vertex in and one vertex in ;
(4) There are exactly two edges of each color in E.
58
THEOREM 1
Two graphs and are partial duals if and only if there exists a partial dual
embedding { } such that for each i, is obtained from by adding an edge
between the vertices of , that are incident with the two edges in E that have the same color,
for each color.
EXAMPLE 2
An example of partial dual embedding
Figure: 1
Where is the disjoint union of two spheres, and .
Following the recipe in the theorem we recover the graphs.
G G{1}
Figure: 2
These graphs are indeed partial duals as they are cores of the following graphs
respectively.
59
Figure: 3
We will now prove theorem:1. The idea behind the proof is construct a
correspondence between partial dual embeddings of ribbon graphs and their (embedded)
cores. It then follows by a theorem that the graphs constructed by this theorem are the cores
of partially dual ribbon graphs.
PROOF
First suppose that and are partial duals, so and are the cores of partially
dual ribbon graphs. Then by theorem: , there exists a partial dual embedding
such that is an arrow-marked ribbon graph describing is
an arrow-marked ribbon graph describing : is the core of ; and is the core of .
A partial dual embedding of graphs { ,E} can be constructed from {
} in the following way: Let be the canonically embedded core of and
let be the canonically embedded core of . Each arrow on meets exactly two vertices of
. For each arrow, add an embedded edge between the two corresponding vertices of
the graph which passes through this arrow. Color the edge with the color of the
arrow that it passes through. The set of edges added in this way forms E.
60
We need to show that is indeed a partial dual embedding of graphs and
the graphs and can be recovered from the partial dual embedding in the way described
by the theorem.
To see that is a partial dual embedding, first note that by construction
and E are all embedded in , and that only the ends of the edges in E meet or .
{ } is a dual embedding since { } is. Since each arrow in meets one
vertex in and one vertex in , each edge in E is incident to vertex in and
one vertex in . The coloring requirement follows since there are exactly two edges of
each color in and the edge colorings of E are induced from .
Finally, can be recovered from by adding edges between the marking
arrows of the same color. Therefore, if u and v are vertices of which are marked with an
arrow of the same color and u and v are vertices of which are marked with an arrow of the
same color and u and v are the corresponding vertices of , then to construct the core of
we need to add an edge between u and v. But since u and v are each incident with the edges in
E of the same color we need to add an edge between the vertices of that are incident with
the two edges in E of the same color. This is exactly the construction described in the
statement of the theorem. Using this for each color gives , completing the proof of
necessity.
Conversely, suppose that { ,E} is a partial dual embedding and that and
are obtained as described in the statement of the theorem. Construct a partial dual
embedding { , } of ribbon graph in the following way: take a small neighbourhood
in of the embedded graph to form ; let =( ); wherever an edge in E
61
meets a boundary of vertices add an arrow pointing in an arbitrary direction which is colored
by the color of the edge in E. is the set of such colored arrows.
To see that { } is a partial dual embedding, note that { } is a dual
embedding since { } is, and that there exactly two arrows of each color since there
are exactly two edges of each color in E.
Let denote the ribbon graph described by the arrow-marked ribbon graph .
Then is the core of (since whenever an edge is added between two vertices of in the
formation of , an edge is added between the corresponding vertices of in the formation
of ). Finally, and are partial dual graphs since, by Theorem: and are partial
dual ribbon graphs.
The corollary below follows from the construction of a partial dual embedding in the
proof above.
COROLLARY 3
If G and are partial duals then the corresponding partial dual embedding as
constructed by Theorem: , is , where . Moreover, G
(respectively ) is obtained from (respectively ) by adding an edge
between the vertices of (respectively ) that are incident with the two edges
in E that have the same color for each color.
DEFINITION
Ifs G and H are partially dual graphs that can be obtained from a partial dual
embedding in the way described by theorem, “Two graphs and are partial
duals if and only if there exists a partial dual embedding { } such that for each i,
is obtained from by adding an edge between the vertices of , that are incident with
62
the two edges in E that have the same color, for each color”, then we say that is
a partial dual embedding for G and H.
THEOREM 4 [EDMOND’S THEOREM]
Two graphs G and H are partial duals if and only if there exists a bijection
, such that
(1) satisfies Edmond’s criteria for some subset
(2) If is incident to an edge in A, and if e is incident to v, then (e)
is incident to a vertex of . Moreover, if both ends of e are incident to v, then the both
ends of (e) are incident to vertices of .
(3) If v (G) is not incident to an edge in A, then there exists a vertex (H)
with the property that (G) is incident to v if and only if (H) is incident to .
Moreover, both ends of E are incident to if and only if both ends of are incident to
.
Here is the subgraph of H induced by the images of the edges from A that are
incident with .
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CHAPTER-5
APPLICATIONS OF DUAL GRAPHS
[10]SECTION-1
GRAPH REPRESENTATIONS
The engineering system is represented by a graph representation; all the reasoning
processes upon the system are substituted by a reasoning of a more mathematical nature over
the graph representation.
For graph representations, the mathematical basis of the duality relation lies in the
duality between linear graphs. By definition, two graphs are dual if set of circuits of one
co-insides with the set of cut-sets of the other. When considering this relation in light of
specific graph representations, duality relations for specific pairs of graph representations are
revealed.
For example, two graph representations were introduced- Flow graph representation
(FGR) and potential graph representation (PGR) (see Table 1). It was then proved that for
each Flow graph representation there exists a corresponding dual Potential graph
representation and vice versa. The duality between the two types of representations did not
imply only that their underlying graphs are dual, but also the vector of flows of the former
representation is equal to the vector of potential differences of the later.
GRAPH REPRESENTATIONS
DEFINITION
The work reported in the paper employs a general approach of associating engineering
domains with general discrete mathematical methods, called Graph representations.
Graph representation is an isomorphic graph-theoretical substitute of an engineering
system, the embedded mathematical knowledge of which is used to map the system’s
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behaviour. Different types of graph representations are characterized by four main parts:
embedded knowledge, relations to other graph representations, represented engineering
domains and rules for construction of the representation.
Till now, several types of graph representations were reported and employed to
represent different engineering domains. We utilize two of the representations: flow graph
representation (FGR) and Potential graph representation (PGR), the basic properties of which
are summarized in Table:1
TABLE 1: GRAPH REPRESENTATIONS
Type of graph
Representation
General
Description
Related
Engineering
disciplines
Example of
Engineering
System
Representation of
the example
Engineering
system
Flow Graph
Representation
(FGR).
Each edge in
FGR is associated
a vector called
‘flow’. Flows in
FGR satisfy the
“flow law”,
stating that sum
of flows in each
cut-set is equal to
zero.
Determine
structures,
static
systems,
electric
circuits.
Potential
Graph
Representation
(PGR).
Each vertex in
PGR is associated
a vector, called
‘potential law’,
saying that the
sum of potential
differences in
each circuit is
equal to zero.
Mechanisms,
gear trains,
electric
circuits.
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[10]SECTION-2
DESIGN THROUGH DUALITY RELATION
Now we introduce a general technique for employing the duality relation between
engineering systems for design and demonstrate it on two practical examples, to obtain a new
engineering design by transferring a known one from some other field through mathematical
relations.
When facing a specific engineering design problem, the important issue to be resolved
prior to commencing a process is to decide what known engineering system from other
engineering domain should be transferred: The problem formulation is transferred from the
domain in which the engineering system is to be found to the second domain. Then it is
checked what known engineering system satisfies the obtained requirements and if such
system is found it is transferred to the original engineering domain. Following is the
algorithmic description of the technique:
THE DUAL GRAPH DESIGN TECHNIQUE
(1) Originally the requirements from the engineering system design are formulated in
the terminology of the relevant engineering domain (original engineering domain).
(2) The problem statement is translated into the terminology of the corresponding graph
representation (original graph representation), and becomes a problem in the
representation.
(3) The problem statement obtained in step 2 translated through the duality relation to
the terminology of the dual graph representation (Secondary graph representations).
(4) The problem statement obtained in step 3 is translated to the terminology of the
second engineering domain that is represented by the dual graph representation.
(5) The problem is solved in the secondary engineering domain.
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(6) The graph of the engineering system obtained in step5 is built. Algorithms for
constructing representations of engineering system are described in Table:1.
(7) The graph representation dual to the graph obtained in step 6 is built the
representation for the original design problem is obtained.
(8) From the graph obtained in step 7, an engineering system from the original
engineering domain is built. The construction process can be performed gradually,
by augmenting one element of the system at a time.
Figure2:presents the flow chart describing above design.
DESIGN BY MEANS OF THE DUALITY RELATION BETWEEN
MECHANISMS AND DETERMINATE TRUSSES
Flow graph representation is used to represent determine trusses and Potential graph
representation is used to represent mechanisms, thus we can establish a knowledge transfer
channel between the two systems passing through duality relation between their
representations.
This channel makes possible designing new trusses, starting from known mechanisms,
or conversely new mechanisms starting from known trusses.
The terms of dual design technique for such a case are listed in Table: 2
TABLE 2: CORRESPONDENCE BETWEEN THE TERMINOLOGY OF
DUAL GRAPH DESIGN TECHNIQUE AND THE CASE STUDY
Dual graph technique Current Example
Original Engineering domain Trusses
Secondary engineering domain Mechanisms
Original graph representation FGR
Secondary (dual) graph representation PGR
The correspondence between the terminologies of the graph representations and the
two engineering domains is briefly described in Table: 3
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TABLE 3: FGR AND PGR CONSTRUCTION RULES AND THE DUALITY
RELATION BETWEEN THEM
Terminology of
the original
engineering domain
(trusses)
Terminology of the
original graph
representation
(FGR)
Terminology in the
secondary
representation
(PGR)
Terminology in
secondary
engineering domain
(mechanisms)
Truss element. (rod,
external force,
reaction).
Edge. Edge. Link, sides.
Area closed by rods. Face. Vertex. Kinematical pair.
Internal force of the
element.
Flow through the
edge.
Potential difference
of the edge.
Relative velocity of
the link.
Cut-set. Circuit.
Following is an example of applying the technique for solution of a specific truss
design problem. Following four steps deal with transferring the problem formulation from
trusses into the terminology of graph representation and then to mechanisms. This transfer
process is schematically outlined in Figure: 3
Step 1: Starting the design problem in the terminology of the original domain.
Step 2: Transferring the design problem into the terminology of the original
graph.
Step 3: Translating the problem to dual graph representation terminology.
Step 4: Translating problem statement from dual graph to terminology of the
secondary engineering domain.
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Figure: 3 The transformation process from the truss to mechanism problem.
Step 5: Solving the problem in the secondary domain. The solution for a mechanism design
problem, as it is stated in step 3, can be obtained in a straightforward manner through
employing instant center method, as shown in Figure :4. Finally, the design of the mechanism
can be translated through the graph representation into a new design of a truss. Steps 5-7 for
obtaining the truss design complying to the original requirements are shown in Figure: 5.
Figure 4: Solution for the mechanism design problem
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Step 6: Constructing the graph for the design solution obtained in the secondary
engineering domain.
Step 7: Constructing graph dual obtained in step 6.
Step 8: Building an engineering system for the original engineering design from the graph
obtained in step 7.
Figure 5: Obtaining a new truss design from the known design of mechanism.
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[13]SECTION-3
AN APPLICATION OF GRAPH THEORY IN
GSM MOBILE PHONE NETWORKS
GRAPH COLORING
In Graph theory, graph coloring is a special case of graph labeling; it is an assignment
of labels traditionally called “colors” to elements of a graph. In its simplest form, it is a way
of coloring the vertices of a graph such that no two adjacent vertices share the same color;
this is called a vertex coloring.
EXAMPLE
Figure: 1
A proper vertex coloring of the graph with 3-colors, the minimum number possible.
The convection of using colors originates from coloring the numbers of a map, where each
face is literally colored. This was generalized to coloring the faces of a graph embedded in
the plane by planar duality it became coloring the vertices and in this form it generalizes to all
graphs.
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THE FOUR COLOR PROBLEM
During the 18th century an interesting coloring problem was dominating the minds of
many mathematicians, called the Four Color Problem. The four color problem or the color
map theorem, states that given any separation of a plane into contiguous regions, called a
map, the regions can be colored using at most four colors so that no two adjacent regions
have the same color. Two regions are called adjacent only if they share a border segment, not
just a point.s
REGION WITH FOUR COLORS:
Figure: 2
For any given map, we can construct its dual graph as follows. Put a vertex inside
each region of the map and connect two distinct vertices by an edge if and only if their
respective regions share a whole segment of their boundaries in common. Then, a proper
vertex coloring of the dual graph yields a proper coloring of the regions of the original map.
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Figure 3: The map of India.
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Figure 4: The dual graph of the map of India.
We use vertex coloring algorithm to find a proper coloring of the map of India with
four colors.
GSM MOBILE PHONE NETWORKS
The Groupe Special Mobile (GSM) was created in 1982 to provide a standard for a
mobile telephone system. The first GSM network was launched in 1991 by Radiolinja in
Finland. Today, GSM is the most popular standard for mobile phones in the world, used by
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over 2 billion people across more than 212 countries. GSM is a cellular network with its
entire geographical range divided into hexagonal cells.
Each cell has a communication tower which connects with mobile phones within the
cell. All mobile phone connect to the GSM network by searching for cells in the immediate
vicinity. GSM networks operate in only four different frequency ranges. The reason why only
four different frequencies suffice is clear: the map of the cellular regions can be properly
colored by using only four different colors!
That is the map of India is colored with a minimum of four colors only. Here regions
sharing the same color to share the same frequency. So, the vertex coloring may be used for
assigning at most four different frequencies for any GSM mobile phone network.
Figure 5: The cells of a GSM mobile phone network.
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CONCLUSION
The dissertation on “A STUDY ON DUAL GRAPHS” in Graph theory deals with a
few interesting topics in dual graphs.
The first chapter covers the Introduction to Graph theory, Basic definitions, examples
and dual graphs.
Theorems on Dual graphs are dealt with in the second chapter.
In the third chapter, a discussion on the self-dual graphs is done.
The fourth chapter deals with the characterization of partially dual graphs.
Applications of Dual graphs are dealt with in the fifth chapter.
BIBLIOGRAPHY
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12. Robin J.Wilson Introduction to Graph Theory, Pearson Education Ltd, Fourth Edition
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ACKNOWLEDGEMENT
I express my heartiest gratitude to the Almighty God, to whom I owe everything whose
continuous inspiration and enlightment made me prepare this dissertation.
I wish to express my sincere thanks to, Principal, Auxilium College (Autonomous),
Vellore-6, for her enthusiastic words and constant encouragement which helped me very much
to carry out this research activity.
I express my sincere and deep sense of gratitude to my guide and Head of the
department of Mathematics , Auxilium College (Autonomous), Vellore-6, for her valuable
guidance and wise
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counsel in bringing out this dissertation.
I offer my humble thanks to all the staff members of the department of Mathematics,
Auxilium College (Autonomous), especially to, Vice Principal (Shift II ), Auxilium College.
I thank my beloved family members especially to my brothers and all well wishers for
their help during the course of my work.
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