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Unformatted text preview: Graphs Lecture 18 CS2110 – Fall 2008 2 Announcements Prelim 2 Tuesday, Nov 18, 7:309pm Uris Auditorium Exam conflicts Email Kelly Patwell ASAP Old exams are available for review on the course website 3 These are not Graphs East West North ...not the kind we mean, anyway 4 These are Graphs K 5 K 3,3 = 5 Applications of Graphs Communication networks Routing and shortest path problems Commodity distribution (flow) Traffic control Resource allocation Geometric modeling ... 6 Graph Definitions A directed graph (or digraph ) is a pair (V, E) where V is a set E is a set of ordered pairs (u,v) where u,v ∈ V Usually require u ≠ v (i.e., no selfloops) An element of V is called a vertex (pl. vertices ) or node An element of E is called an edge or arc V = size of V, often denoted n E = size of E, often denoted m 7 Example Directed Graph (Digraph) V = { a,b,c,d,e,f } E = { (a,b), (a,c), (a,e), (b,c), (b,d), (b,e), (c,d), (c,f), (d,e), (d,f), (e,f) } V = 6, E = 11 b a c d e f 8 Example Undirected Graph An undirected graph is just like a directed graph, except the edges are unordered pairs ( sets ) {u,v} Example: b a c e d f V = { a,b,c,d,e,f } E = { {a,b}, {a,c}, {a,e}, {b,c}, {b,d}, {b,e}, {c,d}, {c,f}, {d,e}, {d,f }, {e,f } } 9 Some Graph Terminology Vertices u and v are called the source and sink of the directed edge (u,v), respectively Vertices u and v are called the endpoints of (u,v) Two vertices are adjacent if they are connected by an edge The outdegree of a vertex u in a directed graph is the number of edges for which u is the source The indegree of a vertex v in a directed graph is the number of edges for which v is the sink The degree of a vertex u in an undirected graph is the number of edges of which u is an endpoint b a c e d f b a c d e f 10 More Graph Terminology A path is a sequence v ,v 1 ,v 2 ,...,v p of vertices such that (v i ,v i+1 ) ∈ E, 0 ≤ i ≤ p – 1 The length of a path is its number of edges In this example, the length is 5 A path is simple if it does not repeat any vertices A cycle is a path v ,v 1 ,v 2 ,...,v p such that v = v p A cycle is simple if it does not repeat any vertices except the first and last A graph is acyclic if it has no cycles A directed acyclic graph is called a dag v v 5 b a c d e f 11 Is This a Dag? Intuition: If it’s a dag, there must be a vertex with indegree zero – why? This idea leads to an algorithm A digraph is a dag if and only if we can iteratively delete indegree0 vertices until the graph disappears b a c d e f 12 Is This a Dag?...
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This note was uploaded on 02/01/2010 for the course CS 2110 at Cornell University (Engineering School).
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