Mathematics - Discrete Mathematics - Paths and Circuits

over 2 years ago (Edited)

Introduction

Hey it's a me again @drifter1!

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...

RESOURCES:

References

Images

https://commons.wikimedia.org/wiki/File:Hamiltonian.png

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

Previous articles of the series

Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
Set Operations → Venn Diagrams, Set Operations, Properties and Laws
Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
Probability → Probability Theory, Probability, Theorems, Example
Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)