1. 5.1  Routing Problems: planning and design of delivery routes.  Euler Circuit Problems: Type of routing problem also known as transversability problem.  Unicursal Tracing: Tracing a diagram without lifting up pencil or retracing any lines.  Closed: end up back where you started.  Open: end up elsewhere Discrete Math

2. 5.2  Graphs: pictures consisting of dots and lines.  Vertices: Dots on graph  Edges: Lines on graph  Loop: Edge that connects the same vertices.  Multiple Edges: More than one edge that connects the same two vertices. Discrete Math

3. 5.3  Graph Concepts  Adjacent vertices: an edge joins them  Adjacent Edges: share a common vertex.  Degree of a vertex: The number of edges at that vertex.  Path: A sequence of adjacent Edges where no edges are repeated.  Circuit: A path that begins and ends at the same vertex.  Connected Graph: There exists a path between every pair of vertices. Discrete Math To be continued…

4. 5.3 (Continued...)  Graph Concepts (Continued...)  Disconnected Graph: There does not exist a path between every pair of vertices.  Bridge: An edge, if erased, creates a graph that is disconnected.  Euler Path: A path that travels each edge once and only once.  Euler Circuit: A circuit that travels each edge once and only once. Discrete Math

5. 5.5  Euler’s Theorem: If a graph contains any odd degrees then it cannot contain a Euler circuit. Every vertex must have an even degree and be connected.  A graph that has only two odd degrees (valences) has an Euler path if it is connected. It must begin and end at the odd vertices.  The sum of the degrees of all the vertices of a graph equals twice the number of edges and therefore is an even number. A graph has an even number of odd vertices (degrees). Discrete Math

6. 5.6  Fleury’s Algorithm for finding Euler circuits:  Make sure the graph is connected and has even degrees.  Pick any vertex as a starting point.  Try not to close yourself off from another part of the graph.  Label the edges in order of travel and put arrows on the edges.  Once you have traveled all the edges you are done. Discrete Math

7. 5.7  Eulerizing Graphs: adding edges to create even vertices.  Add the fewest number of possible edges to create all even degrees.  Eliminate Odd Vertices.  Meaning: these are the edges that will have to be traveled twice.  Semi-eulerizing:Creates an Euler path so two odd vertices are left. Discrete Math To be continued…

8. 5.7 (Continued...)  Eulerizing Graphs: adding edges to create even vertices. (Continued...)  Edge Walker: used to create eulerize rectangular figures.  Walk around the outside of the figure and add an edge to the next vertex every time an odd vertex is found. Once you make your way all the way around, all vertices should be even. Discrete Math