TERM PAPER

OFGRAPH THEORY

&PROBABILITY

SUBMITTED TO SUBMITTED BYMISS MANJIT KAUR PARKASH GUPTA

BCA-MCA(INT) IIIRD SEMESTER 3010070322

Euler's Formula

If G is a planar graph, then any plane drawing of G divides the plane into regions, called faces. One of these faces is unbounded, and is called the infinite face. If f is any face, then the degree of f (denoted by deg f) is the number of edges encountered in a walk around the boundary of the face f. If all faces have the same degree (g, say), the G is face-regular of degree g.

For example, the following graph G has four faces, f4 being the infinite face.

 

 

It is easy to see from above graph that deg f1=3, deg f2=4, deg f3=9, deg f4=8.

 

Note that the sum of all the degrees of the faces is equal to twice the number of edges in the the graph , since each edge either borders two different faces (such as bg, cd, and cf) or occurs twice when walk around a single face (such as ab and gh). The Euler's formula relates the number of vertices, edges and faces of a planar graph. If n, m, and f denote the number of vertices, edges, and faces respectively of a connected planar graph, then we get n-m+f = 2.

The Euler formula tells us that all plane drawings of a connected planar graph have the same number of faces namely, 2+m-n.

 

(Euler's Formula )     Let G be a connected planar graph, and let n , m and f   denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n - m + f = 2 .

 

Proof      We employ mathematical induction on edges, m. The induction is obvious for m=0 since in this case n=1 and f=1. Assume that the result is true for all connected plane graphs with fewer than m edges, where m is greater than or equal to 1, and suppose that G has m edges. If G is a tree, then n=m+1 and f=1 so the desired formula follows. On the other hand, if G is not a tree, let e be a cycle edge of G and consider G-e. The connected plane graph G-e has n vertices, m-1 edges, and f-1 faces so that by the inductive hypothesis,

  n - (m - 1) + (f - 1) = 2

which implies that

n - m + f = 2.

We can obtains a number of useful results using Euler's formula. (A "corollary" is a theorem associated with another theorem from which it can be easily derived.)

Nineteen Proofs of Euler's Formula: V-E+F=2

Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways. Examples of this include the existence of infinitely many prime numbers, the evaluation of zeta(2), the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a formula for testing whether an arithmetic progression contains a square) and the Pythagorean theorem (which according to Wells has at least 367 proofs). This also sometimes happens for unimportant theorems, such as the fact that in any rectangle dissected into smaller

rectangles, if each smaller rectangle has integer width or height, so does the large one.

This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V-E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2.

According to Malkevitch, this formula was discovered in around 1750 by Euler, and first proven by Legendre in 1794. Earlier, Descartes (around 1639) discovered a related polyhedral invariant (the total angular defect) but apparently did not notice the Euler formula itself. Hilton and Pederson provide more references as well as entertaining speculation on Euler's discovery of the formula. Confusingly, other equations such as ei pi = -1 and aphi(n) = 1 (mod n) also go by the name of "Euler's formula"; Euler was a busy man.

The polyhedron formula, of course, can be generalized in many important ways, some using methods described below. One important generalization is to planar graphs. To form a planar graph from a polyhedron, place a light source near one face of the polyhedron, and a plane on the other side.

The shadows of the polyhedron edges form a planar graph, embedded in such a way that the edges are straight line segments.

The faces of the polyhedron correspond to convex polygons that are faces of the embedding. The face nearest the light source corresponds to the outside face of the embedding, which is also convex. Conversely, any planar graph with certain connectivity properties comes from a polyhedron in this way.

Some of the proofs below use only the topology of the planar graph, some use the geometry of its embedding, and some use the three-dimensional geometry of the original polyhedron. Graphs in these proofs will not necessarily be simple: edges may connect a vertex to itself, and two vertices may be connected by multiple edges. Several of the proofs rely on the Jordan curve theorem, which itself has multiple proofs; however these are not generally based on Euler's formula so one can use Jordan curves without fear of circular reasoning.

Proof 1: Interdigitating Trees Proof 2: Induction on Faces Proof 3: Induction on Vertices Proof 4: Induction on Edges Proof 5: Divide and Conquer Proof 6: Electrical Charge Proof 7: Dual Electrical Charge Proof 8: Sum of Angles Proof 9: Spherical Angles Proof 10: Pick's Theorem Proof 11: Ear Decomposition Proof 12: Shelling Proof 13: Triangle Removal Proof 14: Noah's Ark Proof 15: Binary Homology Proof 16: Binary Space Partition Proof 17: Valuations Proof 18: Hyperplane Arrangements Proof 19: Integer-Point Enumeration

History

The Königsberg Bridge problem

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the

history of graph theory.[1] This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy[2] and L'Huillier,[3] and is at the origin of topology.

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees. This study had many implications in theoretical chemistry. The involved techniques mainly concerned the enumeration of graphs having particular properties. Enumerative graph theory then rose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959. Cayley linked his results on trees with the contemporary studies of chemical composition.[4] The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a part of the standard terminology of graph theory. In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature.[5]

One of the most famous and productive problems of graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?". This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year. Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on

surfaces with arbitrary genus. Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory.

The four color problem remained unsolved for more than a century. A proof produced in 1976 by Kenneth Appel and Wolfgang Haken[6][7], which involved checking the properties of 1,936 configurations by computer, was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.[8]

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist Gustav Kirchhoff, who published in 1845 his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.

The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, which has been a fruitful source of graph-theoretic results

Graph Theory and the Bridges of Königsberg

Königsberg was a city in Prussia situated on the Pregel River, which served as the residence of the dukes of Prussia in the 16th century.  (Today, the city is named Kaliningrad, and is a major industrial and commercial center of western Russia.)  The river Pregel flowed through the town, creating an island, as in the following picture.  Seven bridges spanned the various branches of the river, as shown.

The Bridges of Königsberg

A famous problem concerning Königsberg was whether it was possible to take a walk through the town in such a way as to cross over every bridge once, and only once.  An example of a failed attempt to take such a walking tour is shown below.

OK, so this attempt didn't work.  But might there be some other path that would cross every bridge exactly once?  This problem was first solved by the prolific Swiss mathematician Leonhard Euler (pronounced "Oiler"), who invented the branch of mathematics now known as graph theory in the process of his solution.

Graphs

Euler's approach was to regard the spots of land (there are 4 of them) as points to be visited, and the bridges as paths between those points.  The mathematical essentials of the map of Königsberg can

then be reduced to the following diagram, which is an example of what is called a graph:

A graph is a figure consisting of points (called vertices--the plural of vertex) and connecting lines or curves (called edges).  The problem of the bridges of Königsberg can then be reformulated as whether this graph can be traced without tracing any edge more than once.

For each of the vertices of a graph, the order of the vertex is the number of edges at that vertex.  The figure below shows the graph of the Königsberg bridge problem, with the orders of the vertices labeled.

Euler's Solution

Euler's solution to the problem of the Königsberg bridges involved the observation that when a vertex is "visited" in the middle of the process of tracing a graph, there must be an edge coming into the vertex, and another edge leaving it; and so the order of the vertex must be an even number.  This must be true for all but at most two of

the vertices--the one you start at, and the one you end at, and so a connected graph is traversible if and only if it has at most two vertices of odd order.  (Note that the starting and ending vertices may be the same, in which case the order of every vertex must be even.)  Now a quick look at the graph above shows that there are more than two vertices of odd order, and so the graph cannot be traced; that is the desired walking tour of Königsberg is impossible.

Additional Fun with Graphs

1.  Suppose the citizens of Königsberg decided to build an eighth bridge, as in the diagram shown below.  Would a walking tour of Königsberg now be possible?

2.  Show how you could add a ninth bridge to the diagram above, to make the walking tour once again impossible.

A Different Problem with the Same Solution

Euler's solution can also be applied to problems that at first look different from the problem of the Königsberg bridges.  Consider the problem of whether it is possible to draw a continuous curve that passes through each of the ten edges (line segments) of the following figure exactly once.  (A curve that passes through a vertex is not allowed.)

The next figure shows a start on a possible solution.

Is there a systematic way to approach this problem?  To analyze this problem, we will create a graph with four vertices, one for each of the four regions (including the outside region, D).  There will be ten edges in our graph, one for each of the boundary edges between two of the regions.  For instance, our graph will have three edges between the vertices for regions A and D, because there are three boundary edges between regions A and D in the figure above.  The resulting graph can be drawn as follows:

The question of whether there is a continuous curve passing through all ten edges is equivalent to the question of whether this graph can be traced.  Since there are only two vertices of odd order, Euler's theorem not only answers the question in the affirmative, but also tells you that you must start either in region A or in region D.

Can you determine whether it is possible to draw a continuous curve that passes through each of the edges of the following figure exactly once?  Now that you know the secret, you can easily make up your own similar challenges.

Graph Theory Today

Today, graph theory is a highly developed field of mathematics, and is both a fertile ground for the creation of new mathematics and an area with many, many applications.  Many research problems in graph theory are easily stated and easily understood (although perhaps not easily solved).  A few of the applications of graph theory include transportation and warehousing applications, planning and scheduling, analysis of electrical networks, and even understanding the Internet!

 

SUBMITTED TO, SUBMITTED BY, MANJIT KAUR SARABJIT SINGH BCA-MCA (INT) IIIrd SEMESTER ROLLNO.19 3010070056

CONTENTS1. Properties

o 1.1 Possible multiplicity o 1.2 Uniqueness o 1.3 Minimum-cost sub graph o 1.4 Cycle property o 1.5 Cut property

2 Pseudo Code 3 Algorithms 4 MST on complete graphs 5 Related problems 6 See also 7 References 8 External links

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MINI SPANNING TREE

The minimum spanning tree of a planar graph. Each edge is labeled with its weight, which here is roughly proportional to its length.

Given a connected, undirected graph, a spanning tree of that graph is a sub graph which is a tree and connects all the vertices together. A single graph can have many different spanning trees. We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree. A minimum spanning tree or minimum weight spanning tree is then a spanning tree with weight less than or equal to the weight of every other spanning tree. More generally, any undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components.

One example would be a cable TV company laying cable to a new neighborhood. If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths. Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights. A spanning tree for that graph would be a subset of those paths that has no cycles but still connects to every house. There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost.

DEFINATION:-

A tree is a connected graph without cycles.

PROPERTIES OF TREES

° A graph is a tree if and only if there is one and only one path joining any two of its vertices.

° A connected graph is a tree if and only if every one of its edges is a bridge.

° A connected graph is a tree if and only if it has N vertices and N; 1 edges.

DEFINATION

° A sub graph that spans (reaches out to) all vertices of a graph are called a spanning sub graph.

° A sub graph that is a tree and that spans (reaches out to) all vertices of the original graph are called a spanning tree.

° among all the spanning trees of a weighted and connected graph, the one (possibly more) with the least total weight is called a minimum spanning tree (MST).

KRUSKAL'S ALGORITHM Step 1

Find the cheapest edge in the graph (if there is more than one, pick one at random). Mark it with any given color, say red.

Step 2

Find the cheapest unmarked (uncolored) edge in the graph that doesn't close a coloured or red circuit. Mark this edge red.

Step 3

Repeat Step 2 until you reach out to every vertex of the graph (or you have N; 1 coloured edges, where N is the number of Vertices.) The red edges form the desired minimum spanning tree.

o Kruskal Step by Step o Tutorial Kruskal o Interactive Kruskal's Algorithm

PRIM’S ALGORITHM Step 0

Pick any vertex as a starting vertex. (Call it S). Mark it with any given color, say red.

Step 1

Find the nearest neighbor of S (call it P1). Mark both P1 and the edge SP1 red. Cheapest unmarked (uncolored) edge in the graph that doesn't close a coloured circuit. Mark this edge with same color of Step 1.

Step 2

Find the nearest uncolored neighbor to the red sub graph (i.e., the closest vertex to any red vertex). Mark it and the edge connecting the vertex to the red sub graph in red.

Step 3

Repeat Step 2 until all vertices are marked red. The red sub graph is a minimum spanning tree.

Interactive Prim's Algorithm  

Given the graph G below

1. Find a spanning sub graph of G and draw it below.

2. Draw all the different spanning trees of G

3. Of those you had in # 2, which one(s) is (are) minimum spanning trees. (i.e., those those have a minimum sum of their weighted edges.)

Given the weighted graph below:

1. Use Kruskal's algorithm to find a minimum spanning tree and indicate the edges in the graph shown below: Indicate on the edges that are selected the order of their selection.

2. Use Prim's algorithm to find the minimum spanning tree and indicate the edges in the graph shown below. Indicate on the edges that are selected the order of their selection.

PRIM’S ALGORITHMSuppose we are given G = (V, E). We assume G is connected. (If not, thenThe algorithm will find a minimal spanning tree for the component weHappen to start in.)

Let B be the set of tree vertices, initially empty.

Let T be the set of tree edges, initially empty.

Choose any v in V.

Set B = B "UNION" {v}.

While B <> V do

Select the minimum weight edge (u, w) with u in V - B, w in BSet T = T "UNION" {(u. w)}.

Set B = B "UNION" {u}

Example. If G is the graph in Figure 2, then initially choose v = vertex 1. We get the following steps:

Iteration B V - B edge chosen T

1 {1} {2, 3, 4, 5} (1, 5) {(1, 5)}

2 {1, 5} {2, 3, 4} (1, 2) {(1, 5) (1, 2)}

3 {1, 2, 5} {3, 4} (2, 3) {(1, 5), (1, 2), (2, 3)}

4 {1,2,3,5} {4} (3,4) {(1,5),(1,2),(2,3), (3, 4)}

Note that at step 3 we could have chosen to add edge (2, 4) instead of (2, 3).

At step 4 we could have chosen to add edge (2, 4) instead of (3, 4).

Now it is fairly easy to see that the algorithm finds a spanning tree for G. But is it a minimal spanning tree? To see that it is, we need the following lemma.

Lemma. Suppose E1 is a subset of E with the property that E1 is a subset of the edges in a minimal spanning tree T for G. Let V1 be the set of vertices incident with edges in E1. Let (u, v) be an edge of minimal weight with the property that u is in V - V1 and v is in V1. Then E1 union {(u, v)} is also a subset of a minimal spanning tree. (figure3).

Proof. If the edge (u, v) is in the minimal spanning tree T, then we are done. If (u, v) is not in T, on the other hand, then there is a path from u to v in T. Let (x, y) be the edge in this path with exactly one vertex in V1. Call this vertex x. (Figure 4). Let T1 be T with edge (x, y) removed and edge (u, v) added. Then E1 union {(u, v)} is contained in T1 and T1 is a spanning tree. Now by the choice of (u, v) we know that the weight of (u, v) is less than or equal to the weight of (x, y). Therefore the weight of T1 is less than or equal to the weight of T, i.e., T1 is a minimal spanning tree for G.

Kruskal's Algorithm. . Another algorithm for finding a minimum spanning tree uses the set data structure. Let G be a connected graph with n vertices and nonnegative edge weights.

Initialize n components, each one containing one vertex of G.

Now sort the edges in increasing order by weight and set T = the empty set.

Now examine each edge in turn. If an edge joins two components, add it to T and merge the two components into one. If not, discard the edge.

Stop when only one component remains.

Example. Consider the graph in Figure 2.

Sorted edges: (1, 5), (2, 4), (2, 3), (3, 4), (1, 2), (4, 5)

Step Components add T

1 {1}. {2}, {3}, {4}, {5} (1, 5) (1, 5)

2 {1, 5}, {2}, {3}, {4} (2, 4) (1, 5), (2, 4)

3 {1,5},{2,4},{3} (2,3) (1,5),(2,4),(2,3)

4 {1,5},{2,3,4} (1,2) (1,5),(2,4),(2,3),(1,2)

P ROPERTIES

Possible multiplicity

There may be several minimum spanning trees of the same weight; in particular, if all weights are the same, every spanning tree is minimum.

UNIQUENESS

If each edge has a distinct weight then there will only be one, unique minimum spanning tree. The proof of this fact can be done by induction or contradiction. This is true in many realistic situations, such as the cable TV company example above, where it's unlikely any two paths have exactly the same cost. This generalizes to spanning forests as well.

MINIMUM-COST SUB GRAPH

If the weights are non-negative, then a minimum spanning tree is in fact the minimum-cost subgraph connecting all vertices, since subgraphs containing cycles necessarily have more total weight.

CYCLE PROPERTIES

For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of other edges of C, then this edge cannot belong to an MST. Indeed, assume the contrary, i.e., e belongs to an MST T1. If we delete it, T1 will be broken into two subtrees with the two ends of e in different subtrees. The remainder of C reconnects the subtrees, hence there is an edge f of C with ends in different subtrees, i.e., it reconnects the subtrees into a tree T2 with weight less than that of T1, because the weight of f is less than the weight of e.

CUT PROPERTIES

For any cut C in the graph, if the weight of an edge e of C is smaller than the weights of other edges of C, then this edge belongs to all MSTs of the graph. Indeed, assume the contrary, i.e., e does not belong to an MST T1. Then adding e to T1 will produce a cycle, which must have another edge e2 from T1 in the cut C. Replacing e2 with e, would produce a tree T1 of smaller weight.

ALGORITHMS

The first algorithm for finding a minimum spanning tree was developed by Czech scientist Otakar Borůvka in 1926 (see Borůvka's algorithm). Its purpose was an efficient electrical coverage of Moravia. There are now two algorithms commonly used Prim's algorithm and Kruskal's algorithm. All three are greedy algorithms that run in polynomial time, so the problem of finding such trees is in FP, and related decision problems such as determining whether a particular edge is in the MST or determining if the minimum total weight exceeds a certain value are in P. Another greedy algorithm not as commonly used is the reverse-delete algorithm, which is the reverse of Kruskal's algorithm.

The fastest minimum spanning tree algorithm to date was developed by Bernard Chazelle, which is based on the Soft Heap, an approximate priority queue. [1] [2] Its running time is O (e α (e, v)), where e is the number of edges, v is the number of vertices and α is the classical functional inverse of the Ackermann function. The function α grows extremely slowly, so that for all practical purposes it may be considered a constant no greater than 4; thus Chazelle's algorithm takes very close to linear time.

What is the fastest possible algorithm for this problem? That is one of the oldest open questions in computer science. There is clearly a linear lower bound, since we must at least examine all the weights. If the edge weights are integers with a bounded bit length, then deterministic algorithms are known with linear running time. For general weights, there are randomized algorithms whose expected running time is linear.

Whether there exists a deterministic algorithm with linear running time for general weights is still an open question. However, Seth Pettie and Vijaya Ramachandran have found a provably optimal deterministic minimum spanning tree algorithm, the computational complexity of which is unknown.

More recently, research has focused on solving the minimum spanning tree problem in a highly parallelized manner. With a linear number of processors it is possible to solve the problem in O (log n) time. A 2003 paper "Fast Shared-Memory Algorithms for Computing the Minimum Spanning Forest of Sparse Graphs" by David A. Bader and Guojing Cong demonstrates a pragmatic algorithm that can compute MSTs 5 times faster on 8 processors than an optimized sequential algorithm. Typically, parallel algorithms are based on Boruvka's algorithm — Prim's and especially Kruskal's algorithm do not scale as well to additional processors.

Other specialized algorithms have been designed for computing minimum spanning trees of a graph so large that most of it must be stored on disk at all times. These external storage algorithms, for example as described in "Engineering an External Memory Minimum Spanning Tree Algorithm" by Roman Dementiev et al.,[10] can operate as little as 2 to 5 times slower than a traditional in-memory algorithm; they claim that "massive minimum spanning tree problems filling several hard disks can be solved overnight on a PC." They rely on efficient external storage sorting algorithms and on graph contraction techniques for reducing the graph's size efficiently.

MST on complete graphs

It has been shown by J. Michael Steele based on work by Alan M. Frieze that given a complete graph on n vertices, with edge weights chosen from a continuous random distribution f such that f'(0) > 0, as n approaches infinity the size of the MST approaches ζ (3) / f'(0), where ζ is the Riemann zeta function.

For uniform random weights in [0, 1], the exact expected size of the minimum spanning tree has been computed for small complete graphs.

Vertices Expected size Approximate expected size

2 1 / 2 0.5

3 3 / 4 0.75

4 31 / 35 0.8857143

5 893 / 924 0.9664502

6 278 / 273 1.0183151

7 30739 / 29172 1.053716

8 199462271 / 184848378 1.0790588

9 126510063932 / 115228853025 1.0979027

Related problems

A related graph is the k-minimum spanning tree (k-MST) which is the tree that spans some subset of k vertices in the graph with minimum weight.

A set of k-smallest spanning trees is a subset of k spanning trees (out of all possible spanning trees) such that no spanning tree outside the subset has smaller weight. (Note that this problem is unrelated to the k-minimum spanning tree.)

The Euclidean minimum spanning tree is a spanning tree of a graph with edge weights corresponding to the Euclidean distance between vertices.

In the distributed model, where each node is considered a computer and no node knows anything except its own connected links, one can consider Distributed minimum spanning tree. Mathematical definition of the problem is the same but has different approaches for solution.

For directed graphs, the minimum spanning tree problem can be solved in quadratic time using the Chu–Liu/Edmonds algorithm.

REFERENCES

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