Problem StatementWhat is the line covering number for the following graph?SolutionNumber of vertices = |V| = n = 7Line covering number = (α1) ≥ ⌈ n / 2 ⌉ = 3α1 ≥ 3By using 3 edges, we can cover all the vertices.Hence, the line covering number is 3.

Problem StatementWhat is the chromatic number of complete graph Kn?SolutionIn a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number Kn = n.

A graph is traversable if you can draw a path between all the vertices without retracing the same path. Based on this path, there are some categories like Euler’s path and Euler’s circuit which are described in this chapter.Euler’s PathAn Euler’s path contains each edge of ‘G’ exactly once and each vertex of ‘G’ at least once. A connected graph G is said to be traversable if it contains an Euler’s path.ExampleEuler’s Path = d-c-a-b-d-e.Euler’s CircuitIn an Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.ExampleEuler’s Path = a-b-c-d-a-g-f-e-c-a.Euler’s Circuit ... Read More

A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.Graph TheoryDefinition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a  represents an endpoint of an edge. An edge joins two vertices a, b  and is represented by set of vertices it connects.Example − Let us ... Read More

Distance between Two VerticesIt is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.Notation − d(U, V)There can be any number of paths present from one vertex to other. Among those, you need to choose only the shortest one.ExampleTake a look at the following graph −Here, the distance from vertex 'd' to vertex 'e' or simply 'de' is 1 as there is one edge between them. There are many paths from vertex 'd' to vertex ... Read More

It is the number of vertices adjacent to a vertex V. Notation − deg(V). In a simple graph with n number of vertices, the degree of any vertices is − deg(v) = n – 1 ∀ v ∈ G A vertex can form an edge with all other vertices except by itself. So the degree of a vertex will be up to the number of vertices in the graph minus 1. This 1 is for the self-vertex as it cannot form a loop by itself. If there is a loop at any of the vertices, then it is not a ... Read More

Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in Graph Theory. Connectivity defines whether a graph is connected or disconnected.ConnectivityA graph is said to be connected if there is a path between every pair of vertex. From every vertex to any other vertex, there should be some path to traverse. That is called the connectivity of a graph. A graph with multiple disconnected vertices and edges is said to be disconnected.Cut VertexLet 'G' be a connected graph. A vertex V ∈ G ... Read More

Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in Graph Theory. Connectivity defines whether a graph is connected or disconnected. It has subtopics based on edge and vertex, known as edge connectivity and vertex connectivity. Let us discuss them in detail.ConnectivityA graph is said to be connected if there is a path between every pair of vertex. From every vertex to any other vertex, there should be some path to traverse. That is called the connectivity of a graph. A graph with ... Read More

Connected GraphA graph is connected if any two vertices of the graph are connected by a path.Vertex 1Vertex 2PATHaba baca b c, a cada b c d, a c dbcb a c , b ccdc dDisconnected GraphA graph is disconnected if at least two vertices of the graph are not connected by a path. If a graph G is disconnected, then every maximal connected subgraph of G is called a connected component of the graph G.Vertex 1Vertex 2PATHaba bacNot AvailableadNot AvailablebcNot Availablecdc d

Trees are graphs that do not contain even a single cycle. They represent hierarchical structure in a graphical form. Trees belong to the simplest class of graphs. Despite their simplicity, they have a rich structure.Trees provide a range of useful applications as simple as a family tree to as complex as trees in data structures of computer science.TreeA connected acyclic graph is called a tree. In other words, a connected graph with no cycles is called a tree.The edges of a tree are known as branches. Elements of trees are called their nodes. The nodes without child nodes are called ... Read More