Street Networks Ch. 1 Finite Math

Our Learning Goal (why am I doing this?) To be able to figure out the best path to choose when traveling a street network When or how might this come in handy??? Checking parking meters, delivering mail, checking electric meters, etc

Terms that need to be understood for this chapter: Graph Vertices Edges Path Circuit Deadheading finite set of dots & connecting links dots on a graph links connecting vertices on a graph connected sequence of edges (named using vertices) a path that starts and ends at the same vertex covering an edge more than once

so what’s the big deal? (problem) Not only is time valuable, but time could be money $$$ (cha-ching) We often want to end up where we started when we embark on a journey How do we know which direction to go, or which order to make our stops?

Can you trace these edges (destinations) without using the same once twice? Try again, but this time end up at the vertex at which you started.

The Big Deal (solution) Euler Circuit- circuits that cover every edge only once and return you to your starting vertex.

Don’t you hate when a solution just brings up a new problem?!? Can we tell by calculation if a graph has a EC? Or must we try all possibilities? Can we tell by calculation if a graph has a EC? Or must we try all possibilities? We are lucky…this time…

More useful terms: Valence Connected the number of edges meeting at a vertex when every pair of vertices of a graph has at least one path connecting the two vertices.

Is this graph connected? What is the valence number for each vertex? There is not a Euler Circuit for this graph!

Euler Circuit (solution) IF a graph is connected and has all valences even, THEN the graph has an Euler Circuit. * the converse of this is true Now that we can tell if a graph has an EC, How do we know what it is?

Finding the EC 1)Never connect vertices that isolate part of the graph 2)Choose randomly when any option will work 3)Practice

Practice Does this graph have a EC? Show one by numbering the sides with the order that they should be traveled

Practice Does this graph have a EC? Show one by numbering the sides with the order that they should be traveled

Practice from the text Pgs #’s1, 9, 14, 19, 20 Tomorrow: What do we do if there is no EC? Give up??? Tomorrow: What do we do if there is no EC? Give up???

Circuits with reused edges  The Chinese Postman Problem: we must minimize the amount of edges to retrace (Meigu Guan-1962) We’ll keep it simple for now. Assume all edges are of equal length.

The Chinese Postman Solution: 1)Take existing graph and add edges by duplicating existing ones, until you arrive at a graph that it connected & even- valent. 2)Find an EC on the Eulerized graph 3)Squeeze this EC from the Eulerized graph onto the original graph by reusing an edge of the original graph each time the circuit on the Eulerized graph uses an added edge. (Double back on the edge where you added one) 1)Take existing graph and add edges by duplicating existing ones, until you arrive at a graph that it connected & even- valent. 2)Find an EC on the Eulerized graph 3)Squeeze this EC from the Eulerized graph onto the original graph by reusing an edge of the original graph each time the circuit on the Eulerized graph uses an added edge. (Double back on the edge where you added one)

Def: EULERIZING: adding edges to a graph to make all valences even. Don’t forget that adding edges means only duplicates of those that already exist! Eulerize it A B C D C B D A

If a street network is composed of a series of rectangular blocks that forms a large rectangle a certain number of blocks high by a certain number of blocks wide, then the network is called… RECTANGULAR

For rectangular graphs: “edge-walker” ▫Start at corner ▫Walk around edge ▫Approach an odd-valent vertex, then connect it to the next.

Circuits with more complications? All edges are NOT of equal length. (ch. 2) All edges are NOT of equal length. (ch. 2) What about one way streets?!?! Digraph: (directed graph) A graph with limitations or one- ways. Digraph: (directed graph) A graph with limitations or one- ways.

Practice from the text Pgs #’s1, 2, 4-7, 9, 14, 19, 20, 22, 25, 26, 33 Next: What do we do if our destination is a vertex instead of an edge?