Routing Problems

Routing Problems
Euler Circuit Problems
Unicursal Drawings
Graph Theory
Spring 2015
Mathematics in
Management Science
Routing Problems
Examples of routing problems
• Mail/Garbage collection
• Meter reading
• Snow plowing
• Security patrol
Routing Problems
Existence question
Is an actual route possible?
Optimization question
Of all the possible routes, which one
is the optimal route?
Euler Circuit Problem
These are special Routing Problems.
The common thread is so-called
exhaustion requirement;
the route must ‘go everywhere’.
Typically want to begin & end at same
We’ll see that ECPs are easy to solve!
Unicursal Drawings
Want to trace each drawing w/o lifting
pencil or retracing any of the lines.
If we begin & end in the same place,
call it a closed unicursal tracing;
otherwise an open unicursal tracing.
• A graph is a finite set of dots and
connecting links.
• The dots are called vertices.
• The links are called edges.
• The valence of a vertex is the
number edges that meet there.
• Vertices can be even or odd
depending on their valence.
Paths and Circuits
• A path is a finite sequence of
adjacent edges on a graph that joins
two vertices.
• A circuit is a path that begins and
ends at the same vertex.
• A graph is connected if every vertex
can be joined to every other vertex by
a path.
Euler Paths and Circuits
• An Euler path is a path that traverses
each edge exactly once.
• An Euler circuit is a circuit that
traverses each edge exactly once.
Euler Circuit Problem
• Do all graphs have Euler circuits?
Answer: No.
• Can we tell which graphs have an
Euler circuit and which don’t?
Answer: Yes!
First, the graph must be all one
piece--- it must be connected.
Euler’s Circuit Theorem
A graph has an Euler circuit
if and only if
the graph is connected,
all its vertices have even valence.
Any odd vertex means NO Euler
Once we know they exist:
How do we find Euler circuits?