 # Review Euler Graph Theory DEFINITION A NETWORK IS

• Slides: 4 Review Euler Graph Theory: DEFINITION: A NETWORK IS A FIGURE MADE UP OF POINTS (VERTICES) CONNECTED BY NON-INTERSECTING CURVES (ARCS). DEFINITION: A VERTEX IS CALLED ODD IF IT HAS AN ODD NUMBER OF ARCS LEADING TO IT, OTHER WISE IT IS CALLED EVEN. DEFINITION: AN EULER PATH IS A CONTINUOUS PATH THAT PASSES THROUGH EVERY ARC ONCE AND ONLY ONCE. Review cont… v Circuits Circuit- a closed path, with no other repeated vertices than the starting and ending vertices Electrical circuit, an electrical network that has a closed loop giving a return path for the current Paths- a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. Edges- An edge is a line segment that joins two vertices. Vertices- A vertex (plural: vertices) is a point where two or more straight lines meet. Basically it’s a corner. Review cont… Euler’s Circuits v v v A path on a graph that starts and ends of the same vertex, it uses each edge exactly once. He first explored this idea in 1736 with his Seven Bridges of Konigsberg Problem. It is not a Eulerian Circuit unless all of it’s vertices have an even degree. Examples Of Jobs that use Eulerian’s Circuits: v Parker Meters/Meter Maid- The basic premise of an Euler circuit is that you have a set of pathways, lined with parking meters, that need to be traversed. Each pathway begins and ends at a node (a street corner). The goal is to find a way that visits each pathway once, and only once. It's hard work emptying parking meters, & you want to avoid doing any more work than you have to. You can visit each node as many times as you want, but can only walk down each pathway once. You also need to end up where you started. Review cont… v v Hamilton Circuit A circuit that starts at a vertex of a graph, passes through every vertex exactly once, and returns to the starting vertex. The vertices do not need to be of even degree The “Optimal Hamiltonian Path, ” is that in which the least slides are used. Example of Jobs that use Hamilton Circuit: When home developers plan out a new community they use Hamilton’s Circuit to make sure there are multiple ways to exit the community. Another example is going through Corn Mazes.