Undirected Network Summary Notice that the network above has multiple edges joining nodes a to d and the network has a loop at node d. Also c is called an isolated node as it is not connected to any other node. Degree of a Node Every node (or vertex) has a degree, which is the number of paths leaving the node. In the diagram above node a has degree 3, node b has degree 2, node c has degree 0, and node d has degree 5. Note that for node d the loop counts twice towards the degree. Nodes a and d have odd degrees and nodes c and b have even degrees. The sum of the degrees of any network = 2 number of edges in the network. Page 1 of 11
Connected Network (or graph): If there is a path between any two vertices of a graph, then the graph is connected. The graph below is connected as all vertices are connected. The graphs below are disconnected since there is no path connecting A to D. Page 2 of 11
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Planar Graphs A graph that can be drawn without intersecting edges is called a planar graph. Note that at first sight a graph may appear not to be planar; the question is whether it can be redrawn without intersections. If it can then the graph is planar. The graph on the right at first sight appears non planar as the edges intersect. However, it can be redrawn as shown on the right, by stretching one of the edges. Both graphs are clearly planar. Some graphs cannot be transformed into planar graphs, such as the one on the right: Euler s Formula A planar graph divides the plane into a number of regions (sometimes called faces). Remember that the area outside the graph is also a region. The graph shown has 6 regions, numbered 1 to 6. There is an important connection between the number of vertices, number of edges and number of regions. For any connected planar graph: V + R = E + 2 Where V is the number of vertices, R is the number of regions (or faces) and E is the number of edges. For the graph shown V = 4, R = 6, E = 8 So 4 + 6 = 8 + 2 Page 5 of 11
Euler and Hamiltonian Paths Euler Paths and Circuits Undirected Network Summary A network is traversible if it can be drawn or traced without lifting pencil from paper and without retracing any edge. The network must be connected and a path must exist that passes over each edge exactly once. The above describes an Euler path. In an Euler path you trace over every edge exactly once. An Euler circuit is an Euler path which starts and finishes at the same vertex. 1. Euler Path that is also an Euler Circuit If all the vertices of the graph (network) have even degree then an Euler path exists. Also the path forms an Euler Circuit. A circuit occurs when you start and finish at the same vertex (or node). It does not matter which vertex you start with you will always finish at that same vertex. The following are examples of Euler circuits. Note that you are allowed to pass through a vertex more than once. 2. Euler Path An Euler path exists if the graph has exactly two vertices of odd degree. The path can only be traced by starting at one of the vertices of odd degree and finishing at the other vertex of odd degree. You cannot form a circuit. The following are examples of Euler paths. A B You must start at q and finish at t or start at t and finish at q. You must start at A and finish at B or start at B and finish at A You must start at F and finish at A or start at A and finish at F. Page 6 of 11
The following graphs do not form Euler paths or Euler circuits as the above rules do not hold. More than 2 vertices of odd degree. Graph is not connected. More than 2 vertices of odd degree Euler paths are useful for deciding on the most efficient route. Think of a road repair gang or the refuse engineers who empty the garbage bins. They would not want to reverse their trucks back the way they came. This wastes time. They would be interested in a route in which they do not have to back track. They would often prefer to start and end at the same point, traversing a circuit. Hamiltonian Paths and Circuits A Hamiltonian path uses every vertex exactly once. It is important to realise that not all edges need to be used. Hamiltonian Path There are often several Hamiltonian paths in the same graph. The path 1 2 3 4 5 forms a Hamiltonian path as every vertex is visited once only. There are other solutions. Hamiltonian Circuit A Hamiltonian circuit is a Hamiltonian path that starts and ends at the same vertex. The path 1 2 3 5 4 1 forms a Hamiltonian circuit as every vertex is visited once only apart from the starting vertex. There are other solutions. There are no rules for finding Hamiltonian paths or circuits. You can trace them out by inspection. Travelling sales people are often interested in Hamiltonian paths or circuits. They would not want to revisit the same town or area twice. Page 7 of 11
Trees In any tree the number of vertices = number of edges + 1 Minimum Spanning Tree Page 8 of 11
Rather than try to find the minimum spanning tree by trial and error we can use the following steps: Step 1. Choose any vertex and select the smallest edge leading from it. Step 2. Inspect the 2 vertices included so far and select the smallest edge leading from either vertex. If there is a tie, arbitrarily choose any one. Step 3. Inspect all vertices included so far and select the smallest edge leading from any included vertex. If there is a tie, arbitrarily choose any one. Step 4: Repeat step 3 until all vertices in the network are included in the tree. Example 1 1. Choose the vertex A to start with. Inspect the network and find the shortest edge connected to A. This is edge AD. 2. Inspect A and D and find the shortest edge connecting one of these to a third vertex. This is DB 3. Inspect A, D and B and find the shortest edge connecting one of these to another vertex. This is BC. Since all the vertices have been connected this is the minimum spanning tree. The total length of this spanning tree is 9 + 10 + 13 = 32 km. Thus the minimum amount of cable necessary is 32 km, arranged as shown in the last diagram. Page 9 of 11
Example 2 Starting with node a look for the shortest edge connected to a. Repeat the steps to determine the minimum spanning tree as shown below. Notice that there are 2 solutions. Shortest Path The shortest path between two nodes can often be found by inspection. Example 1. By inspection the shortest path between A and F is A-C-E-F = 5 + 8 + 6 = 19 m Page 10 of 11
Example 2. On inspection the shortest path between vertex 1 and vertex 9 is: 1-2-3-6-7-9 which is 4 + 3 + 3 + 2 + 2 = 14 Notice that the shortest path always forms a tree. Page 11 of 11