IB Mathematics Applications & Interpretation Topic AHL 3.16—tree and Cycle Algorithms, Chinese Postman, Travelling Salesman Revision Notes (HL)

IB Mathematics Applications & Interpretation Topic AHL 3.16—tree and Cycle Algorithms, Chinese Postman, Travelling Salesman Revision Notes (HL) | RevisionDojo

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Tree and Cycle Algorithms in Graph Theory

Undirected Graphs and Basic Concepts

In graph theory, an undirected graph is a set of vertices connected by edges, where the edges have no direction.
This forms the basis for many important algorithms in computer science and operations research.

Key concepts include:

Walk: A sequence of vertices and edges, allowing repetition
Trail: A walk with no repeated edges
Path: A walk with no repeated vertices
Circuit: A trail that starts and ends at the same vertex
Cycle: A path that starts and ends at the same vertex, with no repeated vertices except the start and end vertex.

Example

Consider the undirected graph shown in the diagram.

Walk: A sequence of vertices where consecutive vertices are connected by an edge.
- Example: $ A \to B \to C \to D \to B \to E $
Trail: A walk in which no edge is repeated.
- Example: $ A \to B \to C \to D \to E $
Path: A trail in which no vertex is repeated.
- Example: $ A \to B \to D \to E $
Circuit: A trail that starts and ends at the same vertex.
- Example: $ A \to B \to C \to D \to B \to A $
Cycle: A path that starts and ends at the same vertex with no repeated edges or vertices (except the start/end vertex).
- Example: $ A \to B \to C \to D \to A $

Eulerian Trails and Circuits

An Eulerian trail is a trail that uses every edge in a graph exactly once.
An Eulerian circuit is an Eulerian trail that starts and ends at the same vertex.

To determine if an Eulerian trail or circuit exists:

For an Eulerian circuit: All vertices must have even degree.
For an Eulerian trail: Exactly two vertices must have odd degree (these will be the start and end vertices).

Note

The degree of a vertex is the number of edges connected to it.

Hamiltonian Paths and Cycles

A Hamiltonian path is a path that visits each vertex exactly once.
A Hamiltonian cycle is a Hamiltonian path that returns to the starting vertex.

Tip

Unlike Eulerian trails, there's no simple rule to determine if a Hamiltonian path or cycle exists in a general graph. This leads to the famous Traveling Salesman Problem, which we'll discuss later.

Common Mistake

Students often confuse Eulerian and Hamiltonian concepts. Remember: Eulerian deals with using all edges once, while Hamiltonian deals with visiting all vertices once.

Example

Consider the following graph with vertices:
$$V=\{A, B, C, D, E, F\}$$
and edges:
$$E=\{(A, B),(A, C),(B, D),(B, E),(C, D),(C, F),(D, E),(E, F)\}$$
A valid Hamiltonian Path for this graph could be:
$$A \rightarrow B \rightarrow E \rightarrow D \rightarrow C \rightarrow F$$
Each vertex is visited exactly once, but the path does not necessarily return to $\mathbf{A}$.

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Definition

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Note

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Excepteur sint occaecat cupidatat non proident

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Tip

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Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

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Tree and Cycle Algorithms in Graph Theory

Undirected Graphs and Basic Concepts

In graph theory, an undirected graph is a set of vertices connected by edges, where the edges have no direction.
This forms the basis for many important algorithms in computer science and operations research.

Key concepts include:

Walk: A sequence of vertices and edges, allowing repetition
Trail: A walk with no repeated edges
Path: A walk with no repeated vertices
Circuit: A trail that starts and ends at the same vertex
Cycle: A path that starts and ends at the same vertex, with no repeated vertices except the start and end vertex.

Example

Consider the undirected graph shown in the diagram.

Walk: A sequence of vertices where consecutive vertices are connected by an edge.
- Example: $ A \to B \to C \to D \to B \to E $
Trail: A walk in which no edge is repeated.
- Example: $ A \to B \to C \to D \to E $
Path: A trail in which no vertex is repeated.
- Example: $ A \to B \to D \to E $
Circuit: A trail that starts and ends at the same vertex.
- Example: $ A \to B \to C \to D \to B \to A $
Cycle: A path that starts and ends at the same vertex with no repeated edges or vertices (except the start/end vertex).
- Example: $ A \to B \to C \to D \to A $

Eulerian Trails and Circuits

An Eulerian trail is a trail that uses every edge in a graph exactly once.
An Eulerian circuit is an Eulerian trail that starts and ends at the same vertex.

To determine if an Eulerian trail or circuit exists:

For an Eulerian circuit: All vertices must have even degree.
For an Eulerian trail: Exactly two vertices must have odd degree (these will be the start and end vertices).

Note

The degree of a vertex is the number of edges connected to it.

Hamiltonian Paths and Cycles

A Hamiltonian path is a path that visits each vertex exactly once.
A Hamiltonian cycle is a Hamiltonian path that returns to the starting vertex.

Tip

Unlike Eulerian trails, there's no simple rule to determine if a Hamiltonian path or cycle exists in a general graph. This leads to the famous Traveling Salesman Problem, which we'll discuss later.

Common Mistake

Students often confuse Eulerian and Hamiltonian concepts. Remember: Eulerian deals with using all edges once, while Hamiltonian deals with visiting all vertices once.

Example

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

IB Mathematics Applications & Interpretation AHL 3.16—tree and Cycle Algorithms, Chinese Postman, Travelling Salesman Notes

Tree and Cycle Algorithms in Graph Theory

Undirected Graphs and Basic Concepts

Eulerian Trails and Circuits

Hamiltonian Paths and Cycles

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Tree and Cycle Algorithms in Graph Theory

Undirected Graphs and Basic Concepts

Eulerian Trails and Circuits

Hamiltonian Paths and Cycles

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics