Graph Theory - Traversability


Advertisements

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 Path

An 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.

Example

Euler’s Path

Euler’s Path = d-c-a-b-d-e.

Euler’s Circuit

In a Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.

Example

Euler's Circuit

Euler’s Path = a-b-c-d-a-g-f-e-c-a.

Euler’s Circuit Theorem

A connected graph ‘G’ is traversable if and only if the number of vertices with odd degree in G is exactly 2 or 0. A connected graph G can contain an Euler’s path, but not an Euler’s circuit, if it has exactly two vertices with an odd degree.

Note − This Euler path begins with a vertex of odd degree and ends with the other vertex of odd degree.

Example

Euler’s Circuit Theorem

Euler’s Path − b-e-a-b-d-c-a is not an Euler’s circuit, but it is an Euler’s path. Clearly it has exactly 2 odd degree vertices.

Note − In a connected graph G, if the number of vertices with odd degree = 0, then Euler’s circuit exists.

Hamiltonian Graph

A connected graph G is said to be a Hamiltonian graph, if there exists a cycle which contains all the vertices of G.

Every cycle is a circuit but a circuit may contain multiple cycles. Such a cycle is called a Hamiltonian cycle of G.

Hamiltonian Path

A connected graph is said to be Hamiltonian if it contains each vertex of G exactly once. Such a path is called a Hamiltonian path.

Example

Hamiltonian Path

Hamiltonian Path− e-d-b-a-c.

Note

  • Euler’s circuit contains each edge of the graph exactly once.

  • In a Hamiltonian cycle, some edges of the graph can be skipped.

Example

Take a look at the following graph −

Hamiltonian cycle

For the graph shown above −

  • Euler path exists – false

  • Euler circuit exists – false

  • Hamiltonian cycle exists – true

  • Hamiltonian path exists – true

G has four vertices with odd degree, hence it is not traversable. By skipping the internal edges, the graph has a Hamiltonian cycle passing through all the vertices.

Useful Video Courses


Video

Graph Theory

97 Lectures 7 hours

Arnab Chakraborty

Video

Computer Graphics

99 Lectures 6 hours

Arnab Chakraborty

Video

Canva: Become a Graphic Designer

7 Lectures 1 hours

Zach Miller

Video

All Excel Charts and Graphs: Data Visualization in Excel

Featured

31 Lectures 3.5 hours

Abhishek And Pukhraj

Video

Graph Theory Algorithms

32 Lectures 6.5 hours

William Fiset

Video

The Ultimate Canva Graphic Design Course

54 Lectures 4 hours

Sasha Miller

Advertisements