Graph and Tree in Automata Theory

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In computer science, graph theory is one of the most important concepts in every aspect. In automata theory, we represent the state transition diagrams through directed graphs.

In this chapter, we will explore the basics of graphs and trees with examples, different terminologies and applications in detail.

Concept of Graphs

A graph is an abstract representation of a set of objects connected by directed or undirected links. There are two major components of a Graph:

• Vertices − The interconnected objects in a graph.
• Edges − The links that connect some pairs of vertices.

To represent graphs, we can use sets. A graph G is a set of vertices V and edges E, with end vertices associated with each edge.

For instance, the graph in the above figure has 5 vertices (A, B, C, D, and E) and 6 edges (AB, BC, CD, DA, DE, and EA). Now let us see several terminologies of graphs

Graph Terminologies

The following terminologies are frequently used Graph Theory −

• Incident Edges − The edges that meet at a common vertex are known as incident on that vertex which is common.
• Degree of a Vertex − The number of edges incident on a specific vertex is known as degree of the vertex. If the edge is a self-loop, it will be counted twice.
• Isolated Vertex − A vertex of a graph having no incident edge is known as isolated vertex.
• Pendant Vertex − When a vertex of degree one is there in a graph. That is known as pendant vertex or end vertex.

From the above example, it is clear that the graph has edges AE, AB, and AD, with vertex A having a degree of 3. It can be proven that the number of odd degree vertices in a graph is always even.

And in this example, the vertex E has no incident edges so this is isolated vertex, where the C has only one incident edge, this is pendent or end vertex.

Graph Traversal

In a graph, we need to traverse through the graph to get the nodes and their values. Based on the idea there are several terminologies associated to graph traversal. Let us understand one by one.

What is a Walk in Graph Theory?

A walk is a sequence of vertices and edges starting from a vertex, traversing edges, and ending at a vertex. It is a finite alternating sequence, starting and ending with vertices. A walk must not have multiple edges or a vertex traversed.

For example − For the graph above, B(BC)C(CD)D(DF)F is a walk. The label in () represents edge.

There are two types of walks −

• Closed Walk − If we end the walk at the initial starting vertex then that is closed walk.
• Open Walk − If we end the walk at any vertex other than the initial starting vertex then that is open walk.

Circuit in Graph Theory

A circuit is nothing but a walk, more specifically a closed walk, where no vertex appears more than once.

Paths in Graph Theory

A path is a type of open walk, where no vertex appears more than once.

Types of Graphs

We can also classify the graphs into two types −

Connected Graph

Where a graph is a single connected component, then it is regarded as a connected graph.

Disconnected Graph

When multiple disconnected components are present, then it is known as a disconnected graph

Role of Trees in Automata

A tree is an undirected connected graph with no circuit. Trees are a special type of graph with unique properties and applications. Trees (rooted-trees) contains specially designed vertex called a root.

The rest of the nodes in a tree can be partitioned into t disjoint sets (t ≥ 0), where each set represents a tree . These are called sub-trees.

For example, a sample tree is given in the following figure −

Tree Terminologies

In the following table, some tree terminologies are given −

Tree Properties

Let us see some of the important properties of Trees −

• Number of Nodes in a Full Binary Tree − For a full binary tree, the number of nodes is given by the formula: , where h is the height of the tree.
• Number of Nodes in a Complete Binary Tree − For a complete binary tree, the number of nodes will be in range, , where h is the height of the tree.

Applications of Graphs and Trees

The following table highlights the applications of Graphs and Trees in Automata –

Conclusion

In this chapter, we explained the concept of graphs and their special types called trees. We also highlighted their types, properties, important terminologies with formula and diagrams. We also discussed the applications of graphs and trees.