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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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 Spring '08
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 Graph Theory, vertices, Euler Circuits

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