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. July – 2011
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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………………….
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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.
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CONTENTS 1.
Introduction
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2.
Theorems On Dual Graphs
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3.
Self-dual Graphs
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4.
A Characterization Of Partially Dual Graphs 38
5.
Applications of Dual Graphs Conclusion Bibliography
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INTRODUCTION
CHAPTER – I Section – 1 Section - 2 Section – 3
Introduction to Graph theory Basic Definitions and Examples Theorems on Dual graphs
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1 5 10
THEOREMS ON DUAL GRAPHS
CHAPTER – II Section – 1 Section - 2 Section – 3
Theorems On Plane Duality Theorems On Combinatorial Dual Some More Theorems On Duality
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14 17 20
SELF-DUAL GRAPHS
CHAPTER – III Section – 1 Section - 2
Forms Of Self-Duality A Comparison Of Forms Of Self-
24 30
Section – 3
Duality Self-Dual Graphs and Matroids
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A CHARACTERIZATION OF PARTIALLY DUAL GRAPHS
CHAPTER – IV Section – 1 Section - 2 Section – 3
Ribbon Graphs Partial Duality Partial Duality For Graphs
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38 42 48
APPLICATIONS OF DUAL GRAPHS
CHAPTER –V Section – 1 Graph Representations Section - 2 Design Through Duality Relation Section – 3 An Application Of Graph Theory in GSM Mobile Phone Networks
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54 56 61
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 19th 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. 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 hydrocarbons and Percolation theory.
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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 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 CH 4
Propane C3 H 8
Graph theory started with Euler who was asked to find a nice path across the seven Koningsberg bridges.
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The (Eulerian) path should cross over each of the seven bridges exactly once
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.
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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
K1,3
K m,n Figure:3
EDGE CUT For subsets S and S ′ of V denote by [ S , S ′ ] the set of edges with one end in S and the other end in S ′ . An edge cut of G is a E of the form [ S , S ′ ] where S is a non-empty proper subset of V and S ′ =V\S.
BOND OR CUT-SET A minimal non-empty edge cut of G is called a Bond.
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Example
∴ e3 , e4 is a Bond.
Edge cut:{ e3 , e4 } { e1 , e2 , e5 } 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 E1 and E2 such that G ( E1 ) and G ( E2 ) 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 G′ 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 G1 and its dual G1 *. Also consider the graph G2 and its dual G2 * (see Figure: 4). (2) Observe that graph G1 and G2 are two different planar representations of a same graph (say, G). (3) The graph G2 * contains a vertex of degree of degree 5, and the graph G1 * contains no vertex of degree 5. Therefore, G1 * and G2 * are non -isomorphic. So, we have that G1 ≅ G2 but G1 * ≅ G2 *. From (3), we may conclude that two isomorphic planar graphs may have distinct non- isomorphic duals.
G1*
G1
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G2*
G2 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 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 G% 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 G%. 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 G%of G, the line e of G%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 f1 and f 2 . Deleting e from G results in a amalgamation of f1 and f 2 into a single face f (see Figure: 1). Any face of G that is adjacent to f1 or f 2 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 f1 * and f 2 * of G* which correspond to the faces f1 and f 2 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 f1 * an f 2 * 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 G * , b) G\e and G * \ e*
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Dually, we have the following proposition.
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 G′ and a path P such that G′ ∩ 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 Ge(minus isolated vertices)”, H=G* /E(P*) is a combinatorial dual of G′ . By the induction hypothesis, G′ and H can be represented as a geometric dual pair, and G′ is also a combinatorial dual of H. If e1 , e2 are two edges of P, then e1 *, e2 * 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 e2 . But, this is impossible since any cycle in G containing e1 also contains e2 . Hence, all edges of E (P)* are parallel in G* and join two vertices z1 , z2 say, in G*. Let z0 be the vertex in H which corresponds to z1 , z2 . The edges in H incident with z0 form a minimal cut in H. Let C be the corresponding cycle in G′ . As E(C)* separates z0 from H- z0 in H, C is a simple closed curve separating z0 from H- z0 . In particular, C is facial in G′ . Let C1 , C2 be the two cycles in CUP containing P such that E ( Ci )* is the minimal cut consisting of the edges incident with zi , for i=1,2. Now we draw P inside the face F of G′
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bounded by C and represent zi inside Ci for i=1,2. This way we obtain a representation of G* as a geometric dual of G.
PROPOSITION 3 Let G be a 2-connected multigraph and let G* be its combinatorial dual. Then G* is 3connected 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 = G1 ∪ G2 where G1 ∩ G2 consists of two vertices, E( G1 ) ∩ E( G2 ) = φ , and each of G1 , G2 contains at least three vertices. By Theorem 2, G is planar. Then G has a facial cycle C such that C ∩ Gi is path Pi 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 G1 and G2 to be duals of each other is as follows. There is a one-to-one correspondence between the edges in G1 and the edges in G2 such that a set of edges in G1 forms a circuit if and only if the corresponding set in G2 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 Γ in G. Clearly, Γ 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 Γ and the other outside. In other words, the set of edges Γ * in G* corresponding to the set Γ in G is a cut-set in G*. (No proper subset of Γ * 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 G′ be the graph for which there is a one-to-one correspondence between the cut-sets of G and circuits of G′ , and viceversa. Let G* be a dual graph of G. There is a one-to-one correspondence between the circuits of G′ 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 G′ and G*, implying that G′ 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 2isomorphic to a dual of G is also a dual of G”, 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 nonplanar graph does not have a dual. Let G be a non-planar graph. Then G contains K 5 or K 3,3 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
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show that neither K 5 nor K 3,3 has a dual, we have proved the theorem. This we shall prove by contradiction as follows: (a) Suppose that K 3,3 has a dual D. Observe that the cut-sets in K 3,3 correspond to circuits in D and vice versa, since K 3,3 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 K 3,3 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 K 3,3 has nine edges and so must its dual. Thus K 3,3 cannot have a dual. Likewise, (b) Suppose that the graph K 5 has a dual H. Note that K 5 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 K 4 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 f1 and f 2 , we draw a dual edge e* connecting the vertices f1 * and f 2 * 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*). 33
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, F′ ).
(V, E, F)
(V, E, F′ )
(F*, E*, V*)
*
(F′ *, E*, V*)
*
Figure:2 Such a move is called Whitney flip, and the duals of (V, E, F) and (V, E, F ′ ) 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 δ : (V, E, F) → (F*, E*, V*) preserving incidences. A weaker notion requires only a graph isomorphism δ : (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. 34
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( Gˆ ) such that P is a circuit of G if and only if g(p) is a minimal edge-cut of Gˆ . Given a graph G =(V,E), an algebraic dual of G is a graph Gˆ for which there exist an algebraic duality g: E(G) → E( Gˆ ).
(a)
(b)
(c)
(d)
Figure 3: A graph and several of its embeddings.
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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. 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 Gˆ 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 E 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, V - E + F = 2-2k”. ⇒ E = 2-2k- V - F = 2(1-k- V ).
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THEOREM 2 The complete graph K n 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 S n ( w −1) +1 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 Wn 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 W3 . See Figure: 4 for W6 .
Fig: 4 The 6-Wheel and its dual
MATROIDS
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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 M = ( S , I ) is a matroid if: 1. φ ∈ I ; 2. If J ′ ⊆ J ∈ I , then J ′ ∈ I ; and 3. For all A ⊆ S, all maximal independent subsets of A have the same cardinality. An isomorphism between two matroids M 1 = ( S1 , I1 ) and M 2 = ( S 2 , I 2 ) is a bijection
χ : S1 → S2 such that I ∈ I1 if and only if χ (I) ∈ I 2 . If such a χ exists, then M 1 and M 2 are isomorphic denoted M 1 ≅ M 2 Given a graph G = (V, E), the cycle matroid M (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 M is graphic if there exists a graph G such that M = M (G). For a matroid M = (S,I) the dual matroid M *= (S,I*) has ground set S and I ⊆ S in I* if there is a maximal independent set B in M such that I ⊆ S\B. A matroid M is co-graphic if M * is graphic. It is easily shown that if G is a connected planar graph, then M * (G) = M (G*). It is well known that G is algebraically self-dual if and only if cycle matroids of G and G* are isomorphic.
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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.
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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 ( V ′, E ′, F ′ ) such that M(E) ≅ M( E ′ ) 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.
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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 ( V ′, E ′, F ′ ) such that M( E ) ≅ M( E ′ ) is not map self- dual.
PROOF Consider the 3-connected self-dual maps in Figure: 6. X 1 has only self-dualities of order 4, two rotations and two flip rotations, while X 2 has only a left-right reflection and a 180° rotation as a self-duality. Form a new map X by gluing two copies of X 2 to X 1 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 X 2 can give map self-duality since every quadrilateral in X 1 has order 4 under any self-duality.
Figure: 6 In particular, self-dual graphs of connectivity less than 3 cannot in general be reembedded as self-dual maps.
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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, M (G) = M 1 ⊕ M 2 ⊕ ⋅⋅⋅⋅⋅⋅ ⊕ M k , and if G* is a planar dual of G, then M(G*) =M(G)* = M 1 * ⊕ M 2 * ⊕ ⋅⋅⋅⋅⋅⋅ ⊕ M k * . If G is a graph self-dual, then there is a bijection δ : M(G) → M(G*) sending cycles to cycles, and so there is a partition π of {1,2,…….k} such that δ : M i → M π ( i ) , 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 M 1 and M 2 , the cycle matroids of two distinct 3-connected self-dual maps X 1 and X 2 whose only self-dualities are the antipodal map. The matroid M 1 ⊕ M 2 is self–dual, but its only map realizations are as the 1-vertex union of X 1 and X 2 , 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.
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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 (V , E , F ′) of G such that (V, E, F) ≅ ( F ′*, E*, V *) . 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 δ : P → P*, where P* denotes the dual of P. we may regard δ 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 δ 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.
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THEOREM 3 Let M be a self-dual connected matroid with 3-block tree T. Let T ′ be the tree obtained from T by deleting all the pendant nodes, and let M ′ be the 2-connected matroid determined by T ′ . Then M ′ 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 δ 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 ( δ ,{δ α } ) of T onto itself, such that for each node α of T, δα : M α → M f (α ) sends cycles of M α to co-cycles of M f (α ) . The restriction of ( δ ,{δ α } ) to T ′ has the same property and so corresponds to a self-dual permutation of M ′ .
THEOREM 4 Suppose M is a self-dual 2-connected matroid with self-dual permutation δ and let e1 ∈ M . Let {e1 , e2 ,......ek } be the orbit of e1 under δ . Suppose one of the following: (1) k is even and M 0 is a 3-connected matroid or a cycle and δ 0 is a matroid automorphism of M 0 fixing an edge e0 . (2) k is odd and M 0 is a 3-connected self-dual matroid with self-dual permutation δ 0 fixing an edge e0 . For i =1, 2,…., k set M 2i +1 = M 0 and M 2i = M 0 * . Let M ′ be the matroid obtained from M by 2-sums with the matroids M i , amalgamating e0 or e0 * in M i with ei .
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Let δ ′ be defined by δ ′ (e) for e ∈ M − {e1 , e2 ,....., ek } , δ ′ : M i − e0 → M i +1 − e0 is induced by * for i =1, 2,…., k and δ ′ = δ 0 : M k → M1 . Then M ′ is a 2-connected self-dual matroid with self-dual permutation δ ′ . Moreover, every 2-connected self-dual matroid and its self-duality is obtained in this manner.
PROOF The fact that this construction gives a 2-connected self-dual matroid follows at once, since to check if δ ′ is a self-duality, it sufficies to check that (δ ′)α sends cycles to co-cycles on each 3-block. The fact that M 0 must be self-dual if K is odd follows by considering that
δ 1k is a self-duality and maps M 0 = M 1 onto itself. To see that all self-dualities arise this way, let δ ′ : M ′ → M ′ be a self-duality, let α be a pendant node of T, and set M 0 = M α . Let M be the self-dual matroid that results from removing from T ( M ′) the K nodes corresponding to the orbit of the node α . δ ′ induces δ : k M → M. Then the desired δ 0 is (δ )α .
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
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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 δ 0 to be the identity, and simply glue in the copies of the maps corresponding to M 0 and M 0 * compatibly to make a self-dual map. In case 2 we must have that M 0 is a self-dual 3-block containing a self-duality fixing e0 , hence it corresponds to a self-dual map and δ 0 must be a reflection or an order 4 rotation fixing e0 , 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.
e
f
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.
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[8]CHAPTER-4
A CHARACTERISATION OF PARTIAL DUAL GRAPHS SECTION-1 RIBBON GRAPHS S. Chmutov recently introduced the concept of the partial dual G A 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 Gv which is Eulerian. That is Gv 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 Gv is counted twice).
RIBBON GRAPHS DEFINITION A ribbon graph G = ( υ (G), ε (G)) is (possibly non-orientable) surface with boundary represent as the union of two sets of topological discs: a set υ (G) of vertices, and set of edges
ε (G) such that
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(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 subribbon 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 r An arrow marked ribbon graph G 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.
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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 ⊂ ε (G). uuuuu r To describe G as an arrow-marked ribbon graph G \ B , 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; uuuuu r color the two arrows with e; and delete the edge e. This gives a marked ribbon graph G \ B . uuuuu r uuuuu r Moreover, the original ribbon graph G can be recovered from G \ B by adding edges of G \ B as prescribed by the marking arrows. Notice that, if G is a ribbon graph and H is any spanning sub-ribbon graph, then there r is an arrow marked ribbon graph of H 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.
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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.
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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 G A 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 A ⊆ ε (G ) . The partial dual G A of G along A is defined below. (Step P1): Give every edge in ε (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.
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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\ Ac by joining the marked paths constructed above by connecting them along the boundaries of G\ Ac in the natural way. (Step P3): This defines a collection of non-interesting, closed curves on the boundary of G\ Ac 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}
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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 Figure: 6 55
Notice that there is a correspondence between the edges G and G A : every edge of G gives rise to exactly two marking arrows of the same color, and one edge of G A is attached between these two arrows. We will denote the resulting natural bijection between the edge sets by φ : ε (G ) → ε (G A ) .
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 = ( υ (G ), ε (G ) ) be a ribbon graph. We can regard G as a punctured surface. By filling in the punctures using a set of discs denoted υ (G*) . We obtain a surface without boundary ∑ . The natural dual (or Euler-Poincare dual) of G is the ribbon graph G* = ( υ (G*), ε (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 = ∑ \ υ (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 ⊆ ε (G ) and Ac = ε (G ) \A. Then G A \ φ ( Ac ) = (G \ Ac ) * .
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PROOF If e ∈ Ac , then the cycles defining the vertices of G A follow the vertices incident with e in G (See Figure: 2). It then follows that we can delete the edges in Ac before or after forming the partial dual and end up with the same ribbon graph. Thus G A \ φ ( Ac ) = (G \ Ac ) A . But, (G \ Ac ) A = (G \ Ac )*. G A \ φ ( Ac ) = (G \ Ac )*.
PARTIAL DUAL EMBEDDINGS DEFINITION A set {G,H, Σ, M } is a partial dual embedding of ribbon graph G and H if i) {G,H, Σ } is a dual embedding; ii) M is a set of disjoint colored arrows marked on the boundaries of the embedded vertices in υ (G ) ∩ υ ( H ) ⊂ Σ 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 { G% , H%, Σ, M } with the property that Σ \ υ ( H%) ∪ M is an arrow-marked ribbon graph describing G, and Σ \ υ (G%) ∪ M 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 ⊆ ε (G) uuuuuur such that G A = H. Then G can described as an arrow-marked ribbon graph G \ Ac , where Ac = ε (A)\A. Let Σ be the surface obtained from G \ Ac by filling in the punctures. Then { uuuuuur G \ Ac , (G \ Ac )*, Σ } forms a natural dual embedding. The arrow markings on G \ Ac induce a
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set of colored arrows on υ (G \ Ac ∩ υ ((G \ Ac )*) with the property that there are exactly two arrows of each color. Denote this induced set of colored arrows by M . Then {G \ Ac , (G \ Ac )*, Σ, M } is a partial dual embedding. Moreover, Σ \ υ ((G \ Ac )*) describes G by construction, and Σ \ υ ((G \ Ac )) clearly describes G A =H if we use the construction of partial duality from the lemma, “Let G be a ribbon graph, A ⊂ ε (G ) and Ac = ε (A)\A. Then the following construction gives G A : uuuuuur (Step P1′ ) : Present G as the arrow-marked ribbon graph G \ Ac . uuuuuur (Step P 2′ ): Take the natural dual of G \ Ac . The marking arrows on G \ Ac induce marking arrows on (G \ Ac )* . (Step P3′ ): G A is the ribbon graph corresponding to the arrow-marked ribbon uuuuuuuuur graph (G \ Ac )* ”. Conversely, suppose that { G% , H%, Σ, M } is a partial dual embedding with the property that Σ \ υ ( H%) ∪ M is an arrow-marked ribbon graph describing G, and Σ \ υ (G%) ∪ M is an arrow marked ribbon graph describing H. Then G% and H% are precisely the naturally dual marked ribbon graphs described in step P 2′ of the construction of partial dual. Here A is the set of edges of G that are also in G′ .
COROLLARY 5 Let G be a ribbon graph and A ⊆ ε (G). Then (1) υ (G A ) = p (G \ Ac ), where Ac = ε (G ) \ A ; (2) p (G A ) = p (G\A); (3) when G is orientable, g( G A )=
1 (2k(G)+e(G)- p (G\ Ac )- p (G\A)). 2
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SECTION-3 PARTIAL DUALITY FOR GRAPHS DEFINITION If G=( υ (G ), ε (G ) ) is a ribbon graph then we can construct a graph G =( υ (G ), ε (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 A ⊆ ε (G). By the notation G A we mean that G A is the core of G A 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{ G% , H%, Σ, Ε } Where Σ is a surface without boundary G% , H%⊂ Σ are embedded graphs and E is a set of colored edges that are embedded in Σ such that (1) Only the ends of each embedded edge in E meet G%∪ H%⊂ Σ ; (2) { G% , H%, Σ } is dual embedding; (3) Each edge in E is incident to one vertex in υ (G%) and one vertex in υ ( H%) ; (4) There are exactly two edges of each color in E. 59
THEOREM 1 Two graphs G1 and G2 are partial duals if and only if there exists a partial dual % Gi is obtained from G% embedding { G% 1 , G2 , Σ, Ε } such that for each i, i by adding an edge between the vertices of G% i , 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 Σ is the disjoint union of two spheres, G% 1 = ({α , β },{1}) and G2 = ({a, b , c},{1}) . Following the recipe in the theorem we recover the graphs.
G{1}
G Figure: 2
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These graphs are indeed partial duals as they are cores of the following graphs respectively.
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 G1 and G2 are partial duals, so G1 and G2 are the cores of partially % dual ribbon graphs. Then by theorem: , there exists a partial dual embedding {G% 1 , G2 , Σ, M } % such that Σ \ υ (G% 2 ) ∪ M is an arrow-marked ribbon graph describing G1 = Σ \ υ (G1 ) ∪ M is an arrow-marked ribbon graph describing G2 : G1 is the core of G1 ; and G2 is the core of G2 . % A partial dual embedding of graphs { G% 1 , G2 , Σ ,E} can be constructed from { % % % % G% 1 , G2 , Σ, M } in the following way: Let G1 be the canonically embedded core of G1 and G2 let be the canonically embedded core of G% 2 . Each arrow on Σ meets exactly two vertices of % G% 1 ∪ G2 . For each arrow, add an embedded edge between the two corresponding vertices of
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% the graph G% 1 ∪ G2 ⊂ Σ 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. % We need to show that [G% 1 , G2 , Σ, Ε} is indeed a partial dual embedding of graphs and the graphs G1 and G2 can be recovered from the partial dual embedding in the way described by the theorem. % To see that [G% 1 , G2 , Σ, Ε} is a partial dual embedding, first note that by construction % % % G% 1 , G2 and E are all embedded in Σ , and that only the ends of the edges in E meet G1 or G2 . % %% { G% 1 , G2 , Σ } is a dual embedding since { G1 , G2 , Σ } is. Since each arrow in M meets one vertex in V (G1 ) and one vertex in V (G2 ) , each edge in E is incident to vertex in V (G% 1 ) and one vertex in V (G% 2 ) . The coloring requirement follows since there are exactly two edges of each color in M and the edge colorings of E are induced from M . Finally, Gi can be recovered from G% i ∪ M by adding edges between the marking arrows of the same color. Therefore, if u and v are vertices of G% i which are marked with an arrow of the same color and u and v are vertices of G% i which are marked with an arrow of the Gi same color and u and v are the corresponding vertices of G% i , 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 G% i 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 Gi , completing the proof of necessity.
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% G1 and Conversely, suppose that { G% 1 , G2 , Σ ,E} is a partial dual embedding and that G2 are obtained as described in the statement of the theorem. Construct a partial dual % embedding { G% 1 , G2 , Σ , M } of ribbon graph in the following way: take a small neighbourhood % % G2 =( Σ \ G% in Σ of the embedded graph G% 1 to form G1 ; let 1 , ε (G1 ) ); wherever an edge in E meets a boundary of vertices add an arrow pointing in an arbitrary direction which is colored by the color of the edge in E. M is the set of such colored arrows. % %% To see that { G% 1, G2 , Σ,M } is a partial dual embedding, note that { G1, G2 , Σ } is a dual % embedding since { G% 1, G2 , Σ } is, and that there exactly two arrows of each color since there are exactly two edges of each color in E. Let Gi denote the ribbon graph described by the arrow-marked ribbon graph G% i ∪M . Then Gi is the core of Gi (since whenever an edge is added between two vertices of G% i in the formation of Gi , an edge is added between the corresponding vertices of G% i in the formation of Gi ). Finally, G1 and G2 are partial dual graphs since, by Theorem: G1 and G2 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 G A are partial duals then the corresponding partial dual embedding as constructed by Theorem: , is {G \ Ac , G A \ φ ( Ac ), Σ, E} , where Ac = ε (G ) \ A . Moreover, G (respectively G A ) is obtained from G \ Ac (respectively G A \ φ ( Ac ) ) by adding an edge
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between the vertices of G \ Ac (respectively G A \ φ ( Ac ) ) 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 {G% , H%, Σ, E} in the way described by theorem, “Two graphs G1 and G2 are partial % duals if and only if there exists a partial dual embedding { G% 1 , G2 , Σ, Ε } such that for each i, % Gi is obtained from G% i by adding an edge between the vertices of Gi , that are incident with the two edges in E that have the same color, for each color”, then we say that {G% , H%, Σ, E} 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
ϕ : ε (G ) → ε ( H ) , such that (1) ϕ \ A : A → ϕ ( A) satisfies Edmond’s criteria for some subset A ⊆ ε (G ); (2) If v ∈υ (G ) is incident to an edge in A, and if e ∈ ε (G ) is incident to v, then ϕ (e) is incident to a vertex of ϕ ( A)v . Moreover, if both ends of e are incident to v, then the both ends of ϕ (e) are incident to vertices of ϕ ( A)v . (3) If v ∈υ (G) is not incident to an edge in A, then there exists a vertex v′ ∈υ (H) with the property that e ∈ ε (G) is incident to v if and only if ϕ (e) ∈ ε (H) is incident to v′ . Moreover, both ends of E are incident to ν if and only if both ends of ϕ (e) are incident to v′ . Here ϕ ( A)v is the subgraph of H induced by the images of the edges from A that are incident with ν .
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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
General
Related
Example of
Representation of
Representation
Description
Engineering
Engineering
the example
disciplines
System
Engineering system
Flow Graph
Each edge in
Determine
Representation
FGR is associated
structures,
(FGR).
a vector called
static
‘flow’. Flows in
systems,
FGR satisfy the
electric
“flow law”,
circuits.
stating that sum of flows in each cut-set is equal to Potential
zero. Each vertex in
Mechanisms,
Graph
PGR is associated
gear
Representation
a vector, called
electric
(PGR).
‘potential law’,
circuits.
trains,
saying that the sum of potential differences in each circuit is equal to zero.
[10]SECTION-2 67
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.
(6)
The graph of the engineering system obtained in step5 is built. Algorithms for constructing representations of engineering system are described in Table:1.
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(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 Original Engineering domain Secondary engineering domain Original graph representation Secondary (dual) graph representation The correspondence between the
Current Example Trusses Mechanisms FGR PGR terminologies of the graph representations and the
two engineering domains is briefly described in Table: 3
TABLE 3: FGR AND PGR CONSTRUCTION RULES AND THE DUALITY RELATION BETWEEN THEM Terminology of
Terminology of the
Terminology in the
Terminology in
the original
original graph
secondary
secondary
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engineering domain
representation
representation
engineering domain
(trusses) Truss element. (rod,
(FGR) Edge.
(PGR) Edge.
(mechanisms) Link, sides.
reaction). Area closed by rods.
Face.
Vertex.
Kinematical pair.
Internal force of the
Flow through the
Potential difference
Relative velocity of
element.
edge. Cut-set.
of the edge. Circuit.
the link.
external force,
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 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.
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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 1. Arumugam.S and Ramachandran Invitation to Graph Theory, Scitech Publications, Edition 2001. 78
2. Balakrishnan.V.K Graph Theory Schaum’s outlines Tata McGraw Hill Edition 2004. 3. Bela Bollabas Modern Graph Theory, Graduate Texts in Mathematics, Springer Verlag. 4. Bhisma Rao.G.S.S Discrete Mathematics and Graph Theory, Scitech Edition2001 5. Bondy.J.A and Murthy.U.S.R Graph Theory with Applications Mac. Millan London, 1976. 6. Brigitte Servatius and Herman Servatius Self-Dual Graphs, Discrete Math., 1996. 7. Douglus B.West Introduction to Graph Theory, Pearson Education Ltd, Edition 2007. 8. Iain Moffatt A Characterization of Partially Dual Graphs, Google. 9. Narsingh Deo Graph Theory with Applications to Engineering and Computer Science Prentice Hall of India Pvt. Ltd 1974. 10. Dr. Offer Shai and Daniel Rubin Design Through Duality, Google. 11. Reinhard Diestel Graph Theory Graduate Texts in Mathematics, Springer Verlag 1991. 12. Robin J.Wilson Introduction to Graph Theory, Pearson Education Ltd, Fourth Edition 2007. 13. Shariefuddin Pirzada and Ashay Dharwadker Applications of Graph Theory, Journal of Korean Society for Industrial and Applied Mathematics, vol.11 2007. 14. Sooryanarayan.B and Ranganath.G.K Graph Theory and its Applications, Chand’s and Company Ltd. Edition 2001.
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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