Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Paths and Circuits.
I highly suggest checking out the previous parts of the series before this one, or at least the previous two which are related to graphs...
So, without further ado, let's get straight into it!
Paths and Circuits
Starting off from a specific vertex of a graph it's possible to move along the edges. The series of edges that is traced or sequence of respective adjacent vertices is known as a walk. Discussing this topic of course only makes sense in the context of connected graphs.
There are various special types of paths that can be traced during a walk:
Simple Path : no edge is repeated
Elementary Path : no vertex is repeated
Circuit or Closed Path : start and end at the same vertex
Euler path : every edge exactly once
Euler circuit : Euler path that starts and ends at the same vertex
Hamilton path : every vertex exactly once
Hamiltonian circuit : Hamilton path that starts and ends at the same vertex
For example, consider the following graph:
Some valid paths are the following:
Euler Paths and Circuits
In the case of the Euler paths and circuits it's easy to check if there's one:
A graph has an Euler circuit when all vertices are of even degree.
A graph has an Euler path when at most two vertices are of odd degree.
Let's not forget to mention that graphs with Euler (or Eulerian) paths are known as Euler (or Eulerian) graphs.
Hamiltonian Paths and Circuits
In the case of Hamiltonian paths and circuits there are no such rules. The problem is even difficult for computers to solve as it is considered NP-complete...
For example, a valid Hamiltonian Path is the following:
Of course, this graph has more Hamiltonian paths and circuits...
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
