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Unformatted text preview: Chapter 11: Graphs and Trees March 23, 2008 Outline 1 11.1 Graphs: An Introduction 2 11.2 Paths and Circuits 3 11.3 Matrix Representations of Graphs 4 11.5 Trees Graphs: Basic Definitions Informally, a graph is a set of points (called vertices ) joined by lines ( edges ). v 1 v 2 v 3 v 5 v 6 e 1 e 2 e 3 e 5 e 7 e 6 e 4 v 4 Figure: Example of a graph Definition (a) A graph is a pair ( V , E ) , where V is a set of points (also called vertices ), and E is a set of edges . (b) Each edge e E is associated with a pair of points from V . If u and v are associated with the edge e they are called the endpoints of e , we often write uv or { u , v } to represent the edge e . Definition An edge is said to connect its two endpoints. Two vertices are adjacent if there is an edge connecting them. An edge is said to be incident on each of its endpoints. A graph with no vertices is called empty , otherwise it is called nonempty . A loop is an edge which joins a vertex to itself (i.e. e = { v , v } ). Two edges which connect the same set of vertices are said to be parallel . (i.e. e 1 = { v 1 , v 2 } , e 2 = { v 1 , v 2 } ). Example Consider our first example of a graph: v 1 v 2 v 3 v 5 v 6 e 1 e 2 e 3 e 5 e 7 e 6 e 4 v 4 (a) What are its set of vertices, its set of edges, and the edgeendpoint function? (b) Find all edges incident on v 1 , all vertices adjacent to v 1 , all loops, all parallel edges, and all isolated vertices. Solution: (a) Vertex set: V = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } Edge set: E = { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } Edgeendpoint function: Edge Endpoint e 1 { v 1 , v 2 } e 2 { v 1 , v 3 } e 3 { v 1 , v 3 } e 4 { v 2 , v 3 } e 5 { v 5 , v 6 } e 6 { v 5 } e 7 { v 6 } (b) All edges incident on v 1 : e 1 , e 2 , e 3 All vertices adjacent to v 1 : v 2 and v 3 All loops: e 6 and e 7 . Parallel edges: e 2 and e 3 Isolated vertices: v 4 Different Representations of the Same Graph It is often possible to draw the same graph in two different ways. To show that two different drawings represent the same graph, we need to find a labelling of vertices and edges which produces the same edgeendpoint function. Example Show that the following two drawings represent the same graph Solution: We can label both drawings as follows: v 3 v 4 v 5 v 2 v 1 v 1 e 1 e 2 e 4 e 5 e 1 e 2 e 3 e 4 e 5 v 2 v 5 v 3 v 4 e 3 and we have to show that they yield the same edgeendpoint function. In both cases, the table for the edgeendpoint function is the same: Edge Endpoints e 1 { v 1 , v 2 } e 2 { v 2 , v 3 } e 3 { v 3 , v 4 } e 4 { v 4 , v 5 } e 5 { v 5 , v 1 } which is what we wanted to show. Directed Graphs In applications, it is often not the case that we can only go from one vertex, say v 1 , to another vertex, e.g. v 2 , but not from v 2 to v 1 ....
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