An Interactive Introduction to Graph Theory

Transcription

1 An Interactive Introduction to Graph Theory An Interactive Introduction to Graph Theory Chris K. Caldwell 1995 This the first of a series of interactive tutorials introducing the basic concepts of graph theory. Most of the pages of these tutorials require that you pass a quiz before continuing to the next. To keep track of your progress we ask that you first register for this course by selecting the [REGISTER] button below (press [help] for more information). After you are registered, you will be able to start this tutorial, moving back and forth in it using the buttons on the bottom of each page. If you are already registered, you may continue where you left off by again pressing the [REGISTER] button and then re-entering your name and password. 10:57:19 AM

2 Edges: Paths, Circuits, and Connectivity Edges: Paths, Circuits, and Connectivity A path is a connected sequence of edges (connecting vertices) in a graph and the length of the path is the number of edges traversed. Below are some examples of paths: (The first has length five, the other three have length four). A circuit is a path which ends at the vertex it begins, so a loop is a circuit of length one. Of the four paths shown above, only the last two are circuits. Finally, a graph is connected if there is a path connecting every pair of vertices. For example, this graph with red vertices and black edges is not connected, in fact it is made of five separate components (five separate connected subgraphs: four vertices and one vertex with four loops). Reflect a moment on these definitions, then test your understanding with the [QUIZ] before moving [FORWARD]. file:///c /Documents%20and%20Settings/Dan%20Beaty/Desktop/intro4.htm3/10/ :18:36 AM

3 Vertices: Adjacency and Degree Vertices: Adjacency and Degree Two vertices are adjacent if they are connected by an edge. In this graph vertex A is adjacent to B, C and D; but not adjacent to E or F. The degree (or valence) of a vertex is the number of edge ends at that vertex. In the graph above the vertices A, B, C, and E have degree 3. Vertices D and F have degree 2. Notice we are counting edge "ends," so a loop (an edge that connects a vertex to itself) add 2 to the degree. The one red vertex in this graph has degree eight. Take the [QUIZ], then move [FORWARD]. file:///c /Documents%20and%20Settings/Dan%20Beaty/Desktop/intro3.htm3/10/ :18:00 AM

4 Definition of Graph Definition of Graph What do these three problems have in common? Yes, they each involved maps, but more importantly all are questions about connectivity. They ask which land masses are connected by bridges, which countries by borders, which cities by roads? The length and exact shape of the bridges, borders and roads do not matter. For example, we could reduce each land mass of Königsberg to a single dot (which we will call a vertex or node), connected by arcs that we will call edges. This type of object (drawn on the left) is called a graph. Compare this graph on the left with the original drawing above, do you see how the graph contains the information we want (what is connected to what) without the extraneous information about size and shape? The Königsberg question is now: can we find a path around this graph that uses each edge exactly once? (We will answer this question in the tutorial on Euler Circuits.) Notice that we are using "graph" to denote a set of vertices possibly connected by edges; this is very different from the "graph of a function" in algebra classes. Here are a few more examples of graphs: Take the [QUIZ] to see if you understand this new concept and then move [FORWARD]. file:///c /Documents%20and%20Settings/Dan%20Beaty/Desktop/intro.htm3/10/ :17:01 AM

5 Planar Graphs Planar Graphs A graph is planar if it can be drawn (on a plane) so that the edges intersect only at the vertices. Note that it does not matter how it is drawn! It is planar if it is possible to draw it on a plane without edges crossing. For example, consider the first five complete graphs again: The first three are obviously planar, as is the fourth since it can be drawn without the 'diagonals' intersecting: But what about the fifth? Below is one attempt to draw it on a plane but one intersection still remains. Get out some paper and see if you can do better, then check you answer by taking the quiz. file:///c /Documents%20and%20Settings/Dan%20Beaty/Desktop/intro5.htm3/10/ :19:28 AM