Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2012 MATH 1300-101
Last updated: October 2012
Comments and suggestions: Email wimr@mscs.mu.edu
Networks from chapter 7
We consider graphs from 2 different perspectives.
- First we look at connected graphs, where all the nodes are connected by
a path.
The edges may or may not be labeled with numbers.
- For unlabeled graphs we look for minimal subgraphs that
span the collection of all nodes.
- We find 2 ways to built such minimal spanning trees.
First method: Remove edges from circuits.
Second method: Restore necessary edges between nodes as long as you strictly
have to.
- Minimal spanning trees have one edge less than they have nodes.
- In case we have a weighted connected graph, we have Kruskal's algorithm
with which to find a cheapest minimal spanning tree.
- Second we consider adding auxiliary nodes (also called junctions).
- Given 3 nodes, we can find a best additional junction called the
Steiner point.
The connections from it to the original nodes make 120 degrees.
- When we have more than 3 origunal nodes, we may add multiple junctions.
The optimal ones are still called Steiner nodes, each with 3 connections to
other nodes or junctions.
The angles between connections from Steiner points are still 120 degrees.
Example Problem(s)
- Recommended problems from Chapter 7 of the book:
11, 15ab
- Recommended problems from Chapter 7 of the book:
19, 21
- Draw 3 nodes at equal distance from one another (so the triangle between
them is equilateral).
Mark where the Steiner point is inside this triangular area.