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11f6643lec26 - CS 6643 F'11 Lec 26 G=(V,E,w is a weigthed...

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Graph Algorithms Definitions • G=(V,E) – V is a set of vertices or nodes – E is a set of edges with each edge connecting two vertices » Edges are denoted as 2-tuples (u,v), where u, v V – G is an undirected graph is edges are unordered 2-tuples – G is a directed graph if edges are ordered 2-tuples Undirected edge (u,v) is incident on vertices u and v – u and v are adjacent to each other – Directed edge (u,v) is incident from u to v, and v is adjacent to u A path from u to v is a sequence <v 0 , v 1 , …, v k > of vertices, where v 0 = u and v k =v and (v i ,v i+1 ) E for 0 i<k Length of the path is then number edges in the path CS 6643 F '11 Lec 26 2 Definitions Path (u,v) – Exists if v is reachable from u – Is simple if all intermediate vertices are distinct – Is a cycle if v 0 = v k G is connected if there is a path from any vertex to any other vertex A complete graph has all possible edges – Every vertex is adjacent to all other vertices A forest is an acyclic graph A tree is a connected acyclic graph A weighted graph has weights for each edge – G=(V,E,w) is a weigthed graph such that w: E-> is a valid function CS 6643 F '11 Lec 26 3 Definitions Adjacency matrix, A=(a i,j

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