Wim Ruitenburg's Fall 2011 MATH 1300-101

Last updated: October 2011
Comments and suggestions: Email wimr@mscs.mu.edu

We consider graphs from 2 different perspectives.

First we look at connected graphs where all the nodes are directly connected by edges, or at least are indirectly connected when following several edges from node to node. The edges may or may not be labeled with numbers.
For unlabeled trees 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.
In case we have a weighted connected graph, we have Kruskal's algorithm with which to find a cheapest minimal spanning tree.

Second we look at nodes (also called junctions) on a surface, and find a cheapest connecting network, if necessary by adding extra junctions.
Extra junctions must be located in special internal locations called Steiner points. This is straightforward with 3 original junctions. This may get more difficult with 4 or more original junctions.