Introduction to Graph Theory

3.1 3.1.1 What is graph? A graph G = (V, E) consists of a set of objects V = {v1, v2, …} called vertices, and another set E = {e1, e2, …} whose elements are called edges. Each edge ek in E is identified with an unordered pair (vi, vj) of vertices. The vertices vi, vj associated with edge ek are called the end vertices of ek. The most common representation of graph is by means of a diagram, in which the vertices are represented as points and each edge as a line segment joining its end vertices. Often this diagram itself is referred to as a graph. Fig 3-1. In the Fig. 3-1 edge e1 having same vertex as both its end vertices is called a self-loop. There may be more than one edge associated with a given pair of vertices, for example e4 and e5 in Fig. 3-1. Such edges are referred to as parallel edges. A graph that has neither self-loop nor parallel edges are called a simple graph, otherwise it is called general graph. It should also be noted that, in drawing a graph, it is immaterial whether the lines are drawn straight or curved, long or short: what is important is the incidence between the edges and vertices. A graph is also called a linear complex, a 1-complex, or a one-dimensional complex. A vertex is also referred to as a node, a junction, a point, 0-cell, or an 0-simplex. Other terms used for an edge are a branch, a line, an element, a 1-cell, an arc, and a 1-simplex. Because of its inherent simplicity, graph theory has a very wide range of applications in engineering, physical, social, and biological sciences, linguistics, and in numerous other areas. A graph can be used to represent almost any physical situation involving discrete objects and a relationship among them.   3.1.2 Finite and Infinite Graphs Although in the definition of a graph neither the vertex set V nor the edge set E need be finite, in most of the theory and almost all applications these sets are finite. A graph with a finite number of vertices as well as a finite number of edges is called a finite graph; otherwise, it is an infinite graph. 3.1.3 Incidence and Degree When a vertex vi is an end vertex of some edge ej, vi and ej are said to be incident with (on or to) each other. In Fig. 3-1, for example, edges e2, e6, and e7 are incident with vertex v4. Two nonparallel edges are said to be adjacent if they are incident on a common vertex. For example, e2 and e7 in Fig. 3-1 are adjacent. Similarly, two vertices are said to be adjacent if they are the end vertices of the same edge. In Fig. 3-1, v4 and v5 are adjacent, but v1 and v4 are not. The number of edges incident on a vertex vi, with self-loops counted twice is called the degree, d(vi), of vertex vi. In Fig. 3-1, for example, d(v1) = d(v3) = d(v4) = 3, d(v2) = 4, and d(v5) = 1. The degree of a vertex is sometimes also referred to as its valency. Since each edge contributes two degrees, the sum of the degrees of all vertices in G is twice the number of edges in G. 3.1.4 Isolated vertex, Pendent vertex, and Null graph A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are vertices with zero degree. Vertex v4 and v7 in Fig. 3-2, for example, are isolated vertices. A vertex of degree one is called a pendent vertex or an end vertex. Vertex v3 in Fig. 3-2 is a pendant vertex. Two adjacent edges are said to be in series if their common vertex is of degree two. In Fig. 3-2, the two edges incident on v1 are in series.   Fig. 3-2 Graph containing isolated vertices, series edges and a pendant vertex. In the definition of a graph G = (V, E), it is possible for the edge set E to be empty. Such a graph, without any edges, is called a null graph. In other words, every vertex in a null graph is an isolated vertex. A null graph of six vertices is shown in Fig. 3-3. Although the edge set E may be empty, the vertex set V must not be empty; otherwise, there is no graph. In other words, by definition, a graph must have at least one vertex.   Fig. 3-3 Null graph of six vertices.

3.2 Matrix Representation of Graphs

Although a pictorial representation of a graph is very convenient for a visual study, other representations are better for computer processing. A matrix is a convenient and useful way of representing a graph to a computer. Matrices lend themselves easily to mechanical manipulations. Besides, many known results of matrix algebra can be readily applied to study the structural properties of graphs from an algebraic point of view. In many applications of graph theory, such as in electrical network analysis and operation research, matrices also turn out to be the natural way of expressing the problem. 3.2.1 Incidence Matrix Let G be a graph with n vertices, e edges, and no self-loops. Define an n by e matrix A =[a_ij], whose n rows correspond to the n vertices and the e columns correspond to the e edges, as follows: The matrix element A_ij = 1, if j^th edge e_j is incident on i^th vertex v_i, and = 0, otherwise. (a) a b c d e f g h

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v1 0 0 0 1 0 1 0 0 v2 0 0 0 0 1 1 1 1 v3 0 0 0 0 0 0 0 1 v4 1 1 1 0 1 0 0 0 v5 0 0 1 1 0 0 1 0

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v6 1 1 0 0 0 0 0 0 (b) Fig. 3-4 Graph and its incidence matrix. Such a matrix A is called the vertex-edge incidence matrix, or simply incidence matrix. Matrix A for a graph G is sometimes also written as A(G). A graph and its incidence matrix are shown in Fig. 3-4. The incidence matrix contains only two elements, 0 and 1. Such a matrix is called a binary matrix or a (0, 1)-matrix. The following observations about the incidence matrix A can readily be made:

Since every edge is incident on exactly two vertices, each column of A has exactly two 1’^s.

The number of 1’^s in each row equals the degree of the corresponding vertex.

A row with all 0’^s, therefore, represents an isolated vertex.

Parallel edges in a graph produce identical columns in its incidence matrix, for example, columns 1 and 2 in Fig. 3-4.

3.3 Trees

The concept of a tree is probably the most important in graph theory, especially for those interested in applications of graphs. A tree is a connected graph without any circuits. The graph in Fig 3-5 for instance, is a tree. It follows immediately from the definition that a tree has to be a simple graph, that is, having neither a self-loop nor parallel edges (because they both form circuits). Fig. 3-5. Tree Trees appear in numerous instances. The genealogy of a family is often represented by means of a tree. A river with its tributaries and sub-tributaries can also be represented by a tree. The sorting of mail according to zip code and the sorting of punched cards are done according to a tree (called decision tree or sorting tree).   3.3.1 Some properties of Trees

1. There is one and only one path between every pair of vertices in a tree, T.
2. A tree with n vertices has n-1 edges.
3. Any connected graph with n vertices and n-1 edges is a tree.
4. A graph is a tree if and only if it is minimally connected.

Therefore a graph with n vertices is called a tree if

1. G is connected and is circuit less, or
2. G is connected and has n-1 edges, or
3. G is circuit less and has n-1 edges, or
4. There is exactly one path between every pair of vertices in G, or
5. G is a minimally connected graph.

Fig. 3-6 Tree of a monotonically increasing sequences in 4,1,13,7,0,2,8,11,3 3.3.2 Pendent Vertices in a Tree It is observed that a tree shown in the Fig. 3-5 has several pendant vertices. A pendant vertex was defined as a vertex of degree one). The reason is that in a tree of n vertices we have n-1 edges, and hence 2(n-1) degrees to be divided among n vertices. Since no vertex can be of zero degree, we must have at least two vertices of degree one in a tree. This makes sense only if n ³ 2. An Application: The following problem is used in teaching computer programming. Given a sequence of integers, no two of which are the same find the largest monotonically increasing subsequence in it. Suppose that the sequence given to us is 4,1,13,7,0,2,8,11,3; it can be represented by a tree in which the vertices (except the start vertex) represent individual numbers in the sequence, and the path from the start vertex to a particular vertex v describes the monotonically increasing subsequence terminating in v. As shown in Fig. 3-6, this sequence contains four longest monotonically increasing subsequences, that is, (4,7,8,11), (1,7,8,11), (1,2,8,11) and (0,2,8,11). Each is of length four. Computer programmers refer to such a tree used in representing data as a data tree. 3.3.3 Rooted and Binary Tree A tree in which one vertex (called the root) is distinguished from all the others is called a rooted tree. For instance, in Fig. 3-6 vertex named start, is distinguished from the rest of the vertices. Hence vertex start can be considered the root of the tree, and so the tree is rooted. Generally, the term tree means trees without any root. However, for emphasis they are sometimes called free trees (or non rooted trees) to differentiate them from the rooted kind.   Fig. 3-6 Tree. Binary Trees: A special class of rooted trees, called binary rooted trees, is of particular interest, since they are extensively used in the study of computer search methods, binary identification problems, and variable-length binary codes. A binary tree is defined as a tree in which there is exactly one vertex of degree two, and each of the remaining vertices of degree one or three. Since the vertex of degree two is distinct from all other vertices, this vertex serves as a root. Thus every binary tree is a rooted tree. 3.3.4 Spanning Trees So far we have discussed the trees when it occurs as a graph by itself. Now we shall study the tree as a subgraph of another graph. A given graph has numerous subgraphs, from e edges, 2^e distinct combinations are possible. Obviously, some of these subgrphs will be trees. Out of these trees we are particularly interested in certain types of trees, called spanning trees. A tree T is said to be a spanning tree of a connected graph G if T is a subgraph of G and T contains all vertices of G. Since the vertices of G are barely hanging together in a spanning tree, it is a sort of skeleton of the original graph G. This is why a spanning tree is sometimes referred to as a skeleton or scaffolding of G. Since spanning trees are the largest trees among all trees in G, it is also quite appropriate to call a spanning tree a maximal tree subgraph or maximal tree of G. Finding a spanning tree of a connected graph G is simple. If G has no circuit, it is its own spanning tree. If G has a circuit, delete an edge from the circuit. This will still leave the graph connected. If there are more circuits, repeat the operation till an edge from the last circuit is deleted, leaving a connected, circuit-free graph that contains all the vertices of G. 3.3.5 Hamiltonian Paths and Circuits Hamiltonian circuit in a connected graph is defined as a closed walk that traverses every vertex of G exactly once, except of course the starting vertex, at which the walk also terminates. A circuit in a connected graph G is said to be Hamiltonian if it includes every vertex of G. Hence a Hamiltonian circuit in a graph of n vertices consists of exactly n edges. Hamiltonian path: If we remove any one edge from a Hamiltonian circuit, we are left with a path. This path is called a Hamiltonian path. Clearly, a Hamiltonian path in a graph G traverses every vertex of G. Since a Hamiltonian path is a subgraph of a Hamiltonian circuit (which in turn is a subgraph of another graph), every graph that has a Hamiltonian circuit also has a Hamiltonian path. There are, however, many graphs with Hamiltonian paths that have no Hamiltonian circuits. The length of a Hamiltonian path in a connected graph of n vertices is n-1. 3.3.5 Traveling-Salesman Problem A problem closely related to the question of Hamiltonian circuits is the Traveling-salesman problem, stated as follows: A salesman is required to visit a number of cities during a trip. Given the distances between the cities, in what order should he travel so as to visit every city precisely once and return home, with the minimum mileage traveled? Representing the cities by vertices and the roads between them by edges, we get a graph. In this graph, with every edge e_i there is associated a real number (the distance in miles, say), w(e_i). Such a graph is called a weighted graph; w(e_i) being the weight of edge e_i. In our problem, if each of the cities has a road to every other city, we have a complete weighted graph. This graph has numerous Hamiltonian circuits, and we are to pick the one that has the smallest sum of distances (or weights). The total number of different (not edge disjoint, of course) Hamiltonian circuits in a complete graph of n vertices can be shown to be (n-1)! / 2. This follows from the fact that starting from any vertex we have n-1 edges to choose from the first vertex, n-2 from the second, n-3 from the third, and so on. These being independent, results with (n-1)! choices. This number is, however, divided by 2, because each Hamiltonian circuit has been counted twice. Theoretically, the problem of the traveling salesman can always be solved by enumerating all (n-1)!/2 Hamiltonian circuits, calculating the distance traveled in each, and then picking the shortest one. However, for a large value of n, the labor involved is too great even for a digital computer. The problem is to prescribe a manageable algorithm for finding the shortest route. No efficient algorithm for problems of arbitrary size has yet been found, although many attempts have been made. Since this problem has applications in operations research, some specific large-scale examples have been worked out. There are also available several heuristic methods of solution that give a route very close to the shortest one, but do not guarantee the shortest. Exercise 3

Draw all simple graphs of one, two, three and four vertices

Name 10 situations that can be represented by means of graphs. Explain what each vertex and edge represent

Draw a connected graph that becomes disconnected when any edge is removed from it

Draw all trees of n labeled vertices for n=1,2,3,4 and 5

Sketch all binary trees with six pendent edges

Sketch all spanning trees of given graphs in this chapter

Write incidence matrix for all the graphs developed

Find the spanning trees for all the graphs developed

Draw a graph which has Hamiltonian path but does not have Hamiltonian circuit

List different paths from vertex1 to vertex n in each graph developed.